There's no fixed answer, but you look for possible connections between one number and its neighbour and see if this connection continues along the series. Another method is to work out the difference between one number and the next along the series. Note these differences and then work out the differences between the differences. Especially look out for a constant difference. You will need at least six numbers in the series because as you work out the differences you reduce the number of numbers you have, and you need to apply what you think is the pattern to all the numbers given in the series to check out whether you have discovered the right pattern. Remember you're looking for consistency. Another possible solution is to see if the position of a number (1, 2, 3,...) is used to build the number, for example, the numbers could be built on the squares of their position (1, 4, 9,...).
If the series has a particular number of numbers like 7, 12, etc., and the question asks you to fill in a missing number, it may be that the number of numbers in the series, including the missing one is a clue: 7 days in a week; 7 colours of the rainbow; 12 months in the year; 12 signs of the Zodiac; etc. The solution in these cases may be that the numbers represent the number of letters in the weekday, colour, number of letters in the month name, etc. you sometimes have to think "outside the box".