As you learn and practice the tricks make sure you check your results by doing multiplication the way you're used to, until the tricks start to become second nature. Checking your results is critically important: the last thing you want to do is learn the tricks incorrectly.
1. Multiplying by 9, or 99, or 999

Multiplying by 9 is really multiplying by 10-1.
So, 9x9 is just 9x(10-1) which is 9x10-9 which is 90-9 or 81.

Let's try a harder example: 46x9 = 46x10-46 = 460-46 = 414.

One more example: 68x9 = 680-68 = 612.

To multiply by 99, you multiply by 100-1.
So, 46x99 = 46x(100-1) = 4600-46 = 4554.

Multiplying by 999 is similar to multiplying by 9 and by 99.
38x999 = 38x(1000-1) = 38000-38 = 37962.

2Multiplying by 5, 25, or 125

Multiplying by 5 is just multiplying by 10 and then dividing by 2. Note: To multiply by 10 just add a 0 to the end of the number.
12x5 = (12x10)/2 = 120/2 = 60.
Another example: 64x5 = 640/2 = 320.
And, 4286x5 = 42860/2 = 21430.
To multiply by 25 you multiply by 100 (just add two 0's to the end of the number) then divide by 4, since 100 = 25x4. Note: to divide by 4 your can just divide by 2 twice, since 2x2 = 4.
64x25 = 6400/4 = 3200/2 = 1600.
58x25 = 5800/4 = 2900/2 = 1450.
To multiply by 125, you multipy by 1000 then divide by 8 since 8x125 = 1000. Notice that 8 = 2x2x2. So, to divide by 1000 add three 0's to the number and divide by 2 three times.
32x125 = 32000/8 = 16000/4 = 8000/2 = 4000.
48x125 = 48000/8 = 24000/4 = 12000/2 = 6000.

3. Multiplying together two numbers that differ by a small even number

This trick only works if you've memorized or can quickly calculate the squares of numbers. If you're able to memorize some squares and use the tricks described later for some kinds of numbers you'll be able to quickly multiply together many pairs of numbers that differ by 2, or 4, or 6.
Let's say you want to calculate 12x14.
When two numbers differ by two their product is always the square of the number in between them minus 1.
12x14 = (13x13)-1 = 168.
16x18 = (17x17)-1 = 288.
99x101 = (100x100)-1 = 10000-1 = 9999
If two numbers differ by 4 then their product is the square of the number in the middle (the average of the two numbers) minus 4.
11x15 = (13x13)-4 = 169-4 = 165.
13x17 = (15x15)-4 = 225-4 = 221.
If the two numbers differ by 6 then their product is the square of their average minus 9.
12x18 = (15x15)-9 = 216.
17x23 = (20x20)-9 = 391.

4. Squaring 2-digit numbers that end in 5

If a number ends in 5 then its square always ends in 25. To get the rest of the product take the left digit and multiply it by one more than itself.
35x35 ends in 25. We get the rest of the product by multiplying 3 by one more than 3. So, 3x4 = 12 and that's the rest of the product. Thus, 35x35 = 1225.
To calculate 65x65, notice that 6x7 = 42 and write down 4225 as the answer.
85x85: Calculate 8x9 = 72 and write down 7225.

5. Multiplying together 2-digit numbers where the first digits are the same and the last digits sum to 10

Let's say you want to multiply 42 by 48. You notice that the first digit is 4 in both cases. You also notice that the other digits, 2 and 8, sum to 10. You can then use this trick: multiply the first digit by one more than itself to get the first part of the answer and multiply the last digits together to get the second (right) part of the answer.
An illustration is in order:
To calculate 42x48: Multiply 4 by 4+1. So, 4x5 = 20. Write down 20.
Multiply together the last digits: 2x8 = 16. Write down 16.
The product of 42 and 48 is thus 2016.
Notice that for this particular example you could also have noticed that 42 and 48 differ by 6 and have applied technique number 4.
Another example: 64x66. 6x7 = 42. 4x6 = 24. The product is 4224.
A final example: 86x84. 8x9 = 72. 6x4 = 24. The product is 7224

6. Squaring other 2-digit numbers

Let's say you want to square 58. Square each digit and write a partial answer. 5x5 = 25. 8x8 = 64. Write down 2564 to start. Then, multiply the two digits of the number you're squaring together, 5x8=40.
Double this product: 40x2=80, then add a 0 to it, getting 800.
Add 800 to 2564 to get 3364.
This is pretty complicated so let's do more examples.
32x32. The first part of the answer comes from squaring 3 and 2.
3x3=9. 2x2 = 4. Write down 0904. Notice the extra zeros. It's important that every square in the partial product have two digits.
Multiply the digits, 2 and 3, together and double the whole thing. 2x3x2 = 12.
Add a zero to get 120. Add 120 to the partial product, 0904, and we get 1024.
56x56. The partial product comes from 5x5 and 6x6. Write down 2536.
5x6x2 = 60. Add a zero to get 600.
56x56 = 2536+600 = 3136.
One more example: 67x67. Write down 3649 as the partial product.
6x7x2 = 42x2 = 84. Add a zero to get 840.
67x67=3649+840 = 4489.

7. Multiplying by doubling and halving

There are cases when you're multiplying two numbers together and one of the numbers is even. In this case you can divide that number by two and multiply the other number by 2. You can do this over and over until you get to multiplication this is easy for you to do.
Let's say you want to multiply 14 by 16. You can do this:
14x16 = 28x8 = 56x4 = 112x2 = 224.
Another example: 12x15 = 6x30 = 6x3 with a 0 at the end so it's 180.
48x17 = 24x34 = 12x68 = 6x136 = 3x272 = 816. (Being able to calculate that 3x27 = 81 in your head is very helpful for this problem.)

8. Multiplying by a power of 2

To multiply a number by 2, 4, 8, 16, 32, or some other power of 2 just keep doubling the product as many times as necessary. If you want to multiply by 16 then double the number 4 times since 16 = 2x2x2x2.
15x16: 15x2 = 30. 30x2 = 60. 60x2 = 120. 120x2 = 240.
23x8: 23x2 = 46. 46x2 = 92. 92x2 = 184.
54x8: 54x2 = 108. 108x2 = 216. 216x2 = 432.

Practice these tricks and you'll get good at solving many different kinds of arithmetic problems in your head, or at least quickly on paper. Half the fun is identifying which trick to use. Sometimes more than one trick will apply and you'll get to choose which one is easiest for a particular problem.

Arithematic trick

1. The 11 Times Trick
We all know the trick when multiplying by ten – add 0 to the end of the number, but did you know there is an equally easy trick for multiplying a two digit number by 11? This is it:
Take the original number and imagine a space between the two digits (in this example we will use 52:
5_2
Now add the two numbers together and put them in the middle:
5_(5+2)_2
That is it – you have the answer: 572.
If the numbers in the middle add up to a 2 digit number, just insert the second number and add 1 to the first:
9_(9+9)_9
(9+1)_8_9
10_8_9
1089 – It works every time.
2. Quick Square
If you need to square a 2 digit number ending in 5, you can do so very easily with this trick. Mulitply the first digit by itself + 1, and put 25 on the end. That is all!
25² = (2x(2+1)) & 25
2 x 3 = 6
625

3. Multiply by 5
Most people memorize the 5 times tables very easily, but when you get in to larger numbers it gets more complex – or does it? This trick is super easy.
Take any number, then divide it by 2 (in other words, halve the number). If the result is whole, add a 0 at the end. If it is not, ignore the remainder and add a 5 at the end. It works everytime:
2682 x 5 = (2682 / 2) & 5 or 0
2682 / 2 = 1341 (whole number so add 0)
13410
Let’s try another:
5887 x 5
2943.5 (fractional number (ignore remainder, add 5)
29435


4. Multiply by 9
This one is simple – to multiple any number between 1 and 9 by 9 hold both hands in front of your face – drop the finger that corresponds to the number you are multiplying (for example 9×3 – drop your third finger) – count the fingers before the dropped finger (in the case of 9×3 it is 2) then count the numbers after (in this case 7) – the answer is 27.

5. Multiply by 4
This is a very simple trick which may appear obvious to some, but to others it is not. The trick is to simply multiply by two, then multiply by two again:
58 x 4 = (58 x 2) + (58 x 2) = (116) + (116) = 232

6. Calculate a Tip

If you need to leave a 15% tip, here is the easy way to do it. Work out 10% (divide the number by 10) – then add that number to half its value and you have your answer:
15% of $25 = (10% of 25) + ((10% of 25) / 2)
$2.50 + $1.25 = $3.75

7. Tough Multiplication
If you have a large number to multiply and one of the numbers is even, you can easily subdivide to get to the answer:
32 x 125, is the same as:
16 x 250 is the same as:
8 x 500 is the same as:
4 x 1000 = 4,000


8. Dividing by 5
Dividing a large number by five is actually very simple. All you do is multiply by 2 and move the decimal point:
195 / 5
Step1: 195 * 2 = 390
Step2: Move the decimal: 39.0 or just 39
2978 / 5
step 1: 2978 * 2 = 5956
Step2: 595.6

9. Subtracting from 1,000
To subtract a large number from 1,000 you can use this basic rule: subtract all but the last number from 9, then subtract the last number from 10:
1000
-648
step1: subtract 6 from 9 = 3
step2: subtract 4 from 9 = 5
step3: subtract 8 from 10 = 2
answer: 352

10. Assorted Multiplication Rules
Multiply by 5: Multiply by 10 and divide by 2.
Multiply by 6: Sometimes multiplying by 3 and then 2 is easy.
Multiply by 9: Multiply by 10 and subtract the original number.
Multiply by 12: Multiply by 10 and add twice the original number.
Multiply by 13: Multiply by 3 and add 10 times original number.
Multiply by 14: Multiply by 7 and then multiply by 2
Multiply by 15: Multiply by 10 and add 5 times the original number, as above.
Multiply by 16: You can double four times, if you want to. Or you can multiply by 8 and then by 2.
Multiply by 17: Multiply by 7 and add 10 times original number.
Multiply by 18: Multiply by 20 and subtract twice the original number (which is obvious from the first step).
Multiply by 19: Multiply by 20 and subtract the original number.
Multiply by 24: Multiply by 8 and then multiply by 3.
Multiply by 27: Multiply by 30 and subtract 3 times the original number (which is obvious from the first step).
Multiply by 45: Multiply by 50 and subtract 5 times the original number (which is obvious from the first step).
Multiply by 90: Multiply by 9 (as above) and put a zero on the right.
Multiply by 98: Multiply by 100 and subtract twice the original number.
Multiply by 99: Multiply by 100 and subtract the original number.
Bonus: Percentages
Yanni in comment 23 gave an excellent tip for working out percentages, so I have taken the liberty of duplicating it here:
Find 7 % of 300. Sound Difficult?
Percents: First of all you need to understand the word “Percent.” The first part is PER , as in 10 tricks per listverse page. PER = FOR EACH. The second part of the word is CENT, as in 100. Like Century = 100 years. 100 CENTS in 1 dollar… etc. Ok… so PERCENT = For Each 100.
So, it follows that 7 PERCENT of 100, is 7. (7 for each hundred, of only 1 hundred).
8 % of 100 = 8. 35.73% of 100 = 35.73
But how is that useful??
Back to the 7% of 300 question. 7% of the first hundred is 7. 7% of 2nd hundred is also 7, and yep, 7% of the 3rd hundred is also 7. So 7+7+7 = 21.
If 8 % of 100 is 8, it follows that 8% of 50 is half of 8 , or 4.
Break down every number that’s asked into questions of 100, if the number is less then 100, then move the decimal point accordingly.

EXAMPLES:
8%200 = ? 8 + 8 = 16.
8%250 = ? 8 + 8 + 4 = 20.
8%25 = 2.0 (Moving the decimal back).
15%300 = 15+15+15 =45.
15%350 = 15+15+15+7.5 = 52.5
Also it’s usefull to know that you can always flip percents, like 3% of 100 is the same as 100% of 3.
35% of 8 is the same as 8% of 35.