I received my W2 from my previous employer and noticed that it reports that I paid over $900 to social security yet only $186 to federal income tax. I can't see how this could be correct. I feel like I remember paying for for income tax during the year but I may not be able to find my paystubs? Am I right in thinking a mistake has been made? If so, what can I do about it?

No, Fed withholding is much less than FICA. If only $186 was withheld for Fed taxes, then its likely you had some paychecks where none was taken out.

If you think a mistake was made ask your employer to recheck it.

Rough calculation: $900 taken for SS means you had about $15K in wages. Depending on your filing status, number of dependents, earned income credit etc it may be that you owe little to no federal income tax. So off hand the amount that was withheld seems OK.

In the future, do yourself a favor and save your pay stubs.