If I am sentenced to 120 days for 3rd dui and do not show up for jail is warrant issued a felony or misdemeanor?

The warrant follows the crime. DUI is a misdemeanor, so will be the warrant. But, it will be a no bail warrant. Get caught and you'll be transported in chains to the court. I would offer to help, but there is nothing I can do, since the case is over and all that the court is going to do is put you in jail when they catch you.

It will be a misdemeanor warrant.

A third offense is a misdemeanor so it is likely a misdemeanor warrant. The severity of the crime will dictate whether or not bail is available if you are arrested, the severity of the offense and the issuing Judge will determine what the bail is (many counties have "bail schedules" ) or not (there are "no bail" warrants which I think is likely in your situation). As long as you are in Alabama it is unlikely that you will be picked up on the warrant, however, DUI has a way of invading your life in Alabama through drivers license complications. Although, you should definitely speak with a DUI attorney in Alabama and possibly the lawyer you used in California to have this question answered in accordance with the facts of your case.

A warrant is a warrant. It will be a bench warrant issued on your misdemeanor case for failing to serve your jail time.

It sounds like you were convicted of a misdemeanor so the warrant would be for a misdemeanor. If you were convicted for a felony it would be a felony warrant. It will be in the system for ever. You may be stopped and held until California tells Alabama that they will not pick you up and bring you back. This could happen a number of times, in my experience, normally at the least convenient time. Do your jail , which would be 60 actual days then return to Alabama.