The motherboard model number is usually printed (in white lettering) somewhere on the board, itself. Below are some images of current, standard motherboards and the most common form-factors (standard sizes).
Mini-ITX
Micro-ATX
ATX
The expansion slots, where one would install a graphics card, in these images, are located in the lower left quadrant of each image. There are two primary types of expansion cards in modern motherboards; PCI (the old and outdated standard) and PCI Express (the current standard). Despite similar names, the two are not compatible with each other. You cannot install a PCI-Express (PCI-E) expansion card into a PCI expansion slot. Conversely, you cannot install a PCI expansion card into a PCI-E expansion slot.
Now, for the most part, any PCI card will fit and work with any PCI expansion slot. However, PCI Express (or PCI-E) cards/expansion slots come in various sizes. From the shortest expansion slot to the longest, there are PCI-Ex1, PCI-Ex4, PCI-Ex8, and PCI-Ex16 expansion slots. You can tell the difference between PCI and PCI-E by the location of the break in the expansion slot. In a PCI-E expansion slot, the break in the slot is on the left side of the expansion slot. In a PCI slot, the break in the expansion slot is on the right hand.
I would note: the Mini-ITX motherboard show above has a single PCI-Ex16 expansion slot. The Micro-ATX motherboard shown above has two PCI-Ex16 expansion slots (the top and bottom slots), one PCI-Ex1 slot (just below the top PCI-Ex16 slot) and one PCI slot (just below the PCI-Ex1 slot). The ATX motherboard shown above has seven expansion slots. In the image, going from top to bottom, the expansion slots are:
1) PCI-Ex1
2) PCI-Ex16
3) PCI
4) PCI-Ex1
5) PCI-Ex16
6) PCI
7) PCI-Ex16
In today's world of add-on cards, you should be looking for PCI-E expansion cards (more current), but if need be, you can install and use PCI cards.
-Wolf sends