Generally speaking, to find a lowest common denominator the first step you would take is to determine if one denominator can be evenly divided into the other. In this case, you'd be looking at the 8 and the 4. Since you can divide 4 into 8 evenly, it's the larger of the two - the lowest common denominator is 8.
I'm not sure how far you need the solution to go, so I'll keep going with it and you can stop wherever you need to.
Since the lowest common denominator is 8, the next step is to set up the problem in a way that you can actually perform the math - actually make the subtraction. So you have to convert the fractions to fractions that have the same value, but have an 8 in the denominator. The first one is already done, but the second is not.
In order to make 1/4 into something with a denominator of 8, you need to multiply both the top and bottom by 2. That will make it 2/8, which has the same value as 1/4. So the problem then becomes:
7/8 - 2/8
If you need to keep going and actually solve it, you then perform the subtraction. Since they have the same denominator, the denominator of the answer will be the same: 8. And the numerator is just 7-2, so the answer is: 5/8.
Hope this helps!