Given the tone of the thread, I'll assume you want to run a server off your home computer, most likely on some broadband connection behind a residential router. If that's true so far, then I'll keep going. If not, well I guess the rest of the post doesn't apply too well. If you don't know if you're behind a router or not, you probably shouldn't be running a server.

When you introduce a router into the mix, you create a hardware firewall that stands between your computer and the internet. Specifically, the IP address of your computer in the context of your home network (LAN IP) is different from the IP address the rest of the world sees (WAN IP). If the world (read: crackers and script kiddies) tries to connect to your WAN IP (they can't connect the LAN IP, it cannot be done, period), they'll get your router, which doesn't much care about whatever they would try to do. Unfortunately, this also means if you run a server of anything, people trying to connect to it will also hit your router instead.

This is where port forwarding comes in. The port is a tag on packets that indicates what application is to receive the data inside. HTTP is on port 80, so any packet tagged with port 80 will be sent to a web server application at the destination. If you ran a web server behind your router, you could tell the router to forward port 80 to the server; any packets coming to the WAN IP and the router tagged with 80 would be sent along to your server. So, to run a server behind a router, you have to forward the port.

What people fail to grasp pretty much all the time about computer security, specifically "omfg haxxors breaking in", is that the port is only one part of the equation. If you forward port 80 on your router to your computer, great, your computer will get any HTTP requests. But, if you are not running a server that listens to port 80, your computer doesn't care what's in that packet; it gets rid of it. There must be some program listening to a port for any cracker to "break in" to your computer, much like you have to be listening to your door to hear a burglar knock (there's no such thing as breaking the window in computing).

Which brings me to the last thing. So you run a Minecraft server on your computer, you forward the ports, and you give people your WAN IP to connect to. What you're wondering is if anyone can use that to break into your computer. That relies entirely on the security of the Minecraft server. If there's a vulnerability in the server program (which is listening on that port), then sure, someone could exploit it and do Bad Things. Knowing whether there's a vulnerability or not is a risk you take when you run an Internet-facing server. I, for one, don't believe there are any exploitable vectors in Minecraft's server.

To that end, no, running a server does not harm your computer. If anything, it opens a very specific angle of attack that someone could use, and only if they were targeting a Minecraft server on Java on whatever operating system you have. It's an exaggeration, but that's shuffling (properly) a deck of cards and pulling out the Ace of Spades on the first try by picking the fifth card from the deck. Every security professional out there will balk at me, but you're just not going to see honest-to-god cracking happening on a residential connection with Minecraft.

Now if all this went over your head, just go build something on our servers.

 from running a server, though people can gain your ip from literally anywhere (irc, visiting a website or minecraft server etc) so you areas safe from hack attempts running a server as you are from just running a server.  the worst that could happen is you getting dos'd (though you really need to piss people off for that to happen :P