That's a very complicated question that turns on the type of signature you're talking about and, to a lesser degree, the state you're in. The two primary laws you need to look at are E-Sign (the Electronic Signatures in Global and National Commerce Act) and the versions of UETA (the Uniform Electronic Transfers Act) in effect in each state that you do business in. Your regulator may also have specific guidance that you need to follow.
The baseline rule, however, is that unless you're dealing with a consumer transaction, certain types of promissory notes, or certain types of enforcement actions (like foreclosure notices), an electronic signature is usually valid just like a paper signature.
Disclaimer: This message is not legal advice and may not represent the views of the institution that I work for.