Greg's answer is correct, but I would add that you should definitely avoid calling a bag a "purse" if a man is carrying it! The word "purse" is strongly associated with women, so most men will be offended if you say they are carrying a purse. A woman's bag is usually only called a "purse" if it's a particular type of fancy handbag designed for carrying women's personal items. (If you do an image search for "purse" you will see what I mean.) If it's a different type of bag (such as a tote bag or backpack-style bag), just call it a "bag," not a purse.

Women can carry a purse/wallet/bag. A wallet is just for money and cards mostly. A purse and a bag can be the same .... or different. Bags *can* be bigger than a purse. Plus a lot do not call it a purse anymore and use a different term, like bag/carry bag/hand bag/shoulder bag or similar Men can carry wallets (money, cards again) or a type of bag like above.