Scan the numbers to see if the answer is readily apparent. Many times it's as simple as "n+2" or something similar. If you can't immediately see the pattern, set up a table to derive the common differences.
Write out the numbers with one space between them, in the order they are given in the problem. In the space between each pair of numbers, write the difference of the two. For example, it the string is 1, 4, 9, 16, 25, 36... your differences would be 3, 5, 7, 9, 11... Continue this until the numbers in the bottom-most row are all the same or you've run out of numbers to find the difference. For example, the next numbers in this sequence would be 2, 2, 2, 2, 2...
Count the number of rows before the row with the common difference, if there is one. If there is, the number of rows before that is the degree of the equation. For instance, in our example, there were 2 rows before the row of 2s. This means that the equation is in the 2nd degree, or quadratic. Thus, it takes the form "y=an^2+bn+c". You can now set up a system of equations or a matrix and solve for the variables.
Examine the diagonals if you do not get a common difference. If the diagonals are the same or have a pattern, then take a diagonal, find the amount that they are being multiplied by to produce the next number, and add 1. For example, if the diagonals are the same, then they are being multiplied by 1. Add 1+1=2. Each number in the sequence is the previous one multiplied by 2. This can also be denoted as a(sub)n=2^n.
Look for the initial row translated a space to the right in the next row in the common difference table. If this happens, then you are dealing with a recursive sequence (one in which you have to do something to the previous term to get the next). If this is the case, you can backwards calculate the next term in the sequence by examining the last number on the first row of the table and the number above and to the left of it. Knowing the basics of common differences and recursive sequences will be adequate for most sequence problems. For more advanced solutions, have a look at the page in the resources section.