I am trying to play XCom Enemy Unknown with my brother but his laptop won't run it, and I can't find a way to open two copies simultaneously on my pc. I could with other games such as borderlands 2, but the same method doesn't work
XCOM does not have a hotseat functionality (2 players, 1 game, 1 computer), and since the PC version runs on steam, you cannot run two instances of the program at once.
Unfortunately, I don't believe there is currently a way to play against your brother with only one computer.
There is a way, but it requires your PC to be relatively powerful and have enough RAM: Virtualize+multiseat the second instance to effectively turn your one powerful PC into a very good one plus a second one with probably acceptable performance. This worked fine for Minecraft (2 players plus bukkit), though for XCOM the virtual PC's performance might force you to play on lower detail settings. Anyway that way you can use a second monitor opposite (no peeking!) and use a separate pair of keyboard/mouse so you can have a true multiplayer experience. Let me know how if you got trouble following Tyler's instructions (the free VMWare player is enough actually, no need to buy anything, and the virtual GPU supports 3D acceleration), and I'd like to hear about whether the performance is ok.
Note that, as Raven Dreamer hinted at, the virtual instance will require you to run Steam from another account with a second bought copy of XCOM, unless multiplayer also works in offline mode.