Paper size in Europe should be a little smaller than A5 so that your card fits easily into a C5 envelope. A5 is 149 x 210 mm. Acceptable size for a card is 145 x 204 mm, give or take.
Paper weight: 300 gsm board (e.g. Artboard duo). Very important! Anything lighter looks unprofessional and cheap.
Finish: The front of the card should be glossy, which is called varnished in printers' terms (e.g. UV varnish). The back of the card can be matt.
Design specs: If you design the card yourself, your artwork needs to be the actual size in mm at 300 dpi. Actual size means finished card size + a little extra, called the bleed - usually around 3 mm. The bleed is where your cards are cut from large sheets. Unless you are good at these things, get help from a pro.