I'm going to change my old ADSL modem. One friend suggested me to buy a router instead of a modem. What's the difference?
Also, I want to be able to browse the web from a Linux system.
Well, both pieces of equipment do different jobs, and they don't overlap. You need the modem to connect to your DSL ISP. So, if the one you have doesn't work, get a new one. If it does work ald all you want to do is share your connection among more than one computer, then maybe you should get a router. The ones made by Netgear and SMC seem to be a bit better than the others, but Linksys and DLink make some inexpensive ones that work fine. (I've got a Linksys BEFSR41 4 port wired router sharing a connection for 3 computers right now, but I'd recommend the DLink over the Linksys).
I'd also suggest that you check into a hybrid product. The phone company that I work for uses an Actiontek DSL gateway, which solves both problems in one piece of equipment. That model seems to be reliable, and its usually retails for about $60USD. Be sure that you ISP will support it, of course. This gateway is a DSL modem with a built-in router (wireless too), so it does both jobs. There are comparable models out there by other manufacturers as well, I'm sure.
BTW, you mentioned that you wanted to connect a Linux machine. That should be absolutely no problem either. I've got a quad-boot XP/Gentoo/Slackware/FreeBSD machine that connects just fine, regardless of which OS I'm in.
The only thing that you want to watch out for is that your router (if you get one) is capable of connecting to your ADSL provider. Do they require that you install a PPPoE client? If so, be sure that your router can connect that way too (most do, just be sure to read the specs before you buy).
Hope this helps.