The standard size for a business card is 90 x 55mm, printed on one or both sides and generally printed on 300gsm+ stock (paper). But nowadays there really isn't a standard business card. Sizes can vary enormously, including square and circle shaped cards. Additional finishes include matt or gloss lamination, which give them extra gloss or a plastic satin finish. There's also a spot UV varnish (this allows you to make a selected area shine), embossing and hot stamp metallic foils. Business cards can even be made from plastic, this is a more costly option but it does help to make your card stand out. Both paper and plastic cards can be used as loyalty or promotional cards as well.
There are so many designs of business cards there are entire books dedicated just to business card design.