Equifax just announced last weekend that a hack compromised the personally identifiable information of over 143 million people. Your credit score is very important when applying for credit cards, mortages, or any sort of loan. Having a bad credit score will lead to higher interest rates or getting your application denied.
Fighting credit score fraud is very difficult. The first step is to even realize that your identity has been compromised. The main companies, Equifax, TransUnion, and Experian, give free credit reports only once a year, and only on request. This means unless you pay a monthly subscription for credit monitoring, it might take a full year or longer to find out your identity has been stolen.
Criminals can apply for government ID in your name, such as a driver's license. With government ID they can do a lot of stuff, such as apply for credit cards, take out loans, or purchase vehicles. They obviously won't pay back those loans, and you'll be on the hook. They could commit crimes with "your" vehicle, and you'll be issued the fines and/or arrest warrants.
If you haven't requested your free credit reports this year, do it now. You can also put a fraud alert on your account which is free and will notify you if someone applies for credit. You can also pay the monthly subscription for credit monitoring services from the abovementioned companies.
If you know for a fact that you have been compromised, you can put a freeze on your credit, which will deny any access to your credit reports.