10/100/1000 are the "modes" regarding speed, in Megabits per second (Mbps and NOT MB/s).
Cat 5 is essentially another name for ethernet.
Your internet speeds are mostly likely only a few mbps. You only really *need* gigabit ethernet (1000 megabits=~1 gigabit) if you have a fiberoptic line with enourmous speeds, or you happen to be on a gigabit network and need to transfer huge files fast.
Since you're connecting this directly to the modem, I think it's safe to assume you are NOT on a gigabit network right now. Since chances are you aren't even hitting the 10 number on your current one, cutting 1000 to 100 shouldn't make much difference (and if you are, you're first thing not going to be above 15, and second thing you're extremely lucky
The world isn't likely about to see internet speeds exceeding 100 for quite a while, and MOST computers have 10/100. If in 10 years we're approaching those speeds as standard, and you still happen to have the same computer, you can always add a PCI card.