Torrent not downloading? Download seems to be stuck and making no progress? The first thing to check is that the download is actually running and not waiting in the queue to start. Read about Vuze's queueing system.
OK, so it really is downloading, next up to check is that the download has sufficient availability for you to download. Take a look at the overview of 'availability' - an availability of 0.0 means that nobody *you are connected to yet* in the swarm has any of the download's files available, for example. Poor availability isn't necessarily terminal, there may be people out there with the files in the swarm who you will eventually connect to - be patient especially at the start of a download. If you continue to be connected to 0 peers/seeds check the tracker status to see if the tracker is behaving - right-click on the download, select 'show details' and then click on the 'Sources' tab to show the status of the various peer sources. Look to see if there are any error status messages against trackers.
OK, there appear to be lots of seeds, why am I not connected to any of them?
A good way to get a feel for the swarm is to go to the peers view - this shows you details of all the peers you are connected to in the swarm and you can use it to see if other peers are successfully downloading while you are not. If you see lots of peers with an availability that is the same as your availability then it means that the swarm is somewhat choked, perhaps due to a superseed. If everyone is stuck at the same availability and not downloading, and there appear to be a lot of seeds that are not sending anyone any data, then the download is likely bad.
The peers that will send you data are the ones that aren't choking you (check the 'Choked by the peer' column for entries with no '*' in them) and to which Vuze has one or more outstanding requests (check the 'Out Reqs' column) - the 'Download Speed' column shows how quickly the peer is sending you data.
If there are large numbers of peers that have outstanding requests and are sending you no data you can try selecting them, right-clicking and then 'kick and ban' (in general this is not a good thing to do, but it is worth experimenting; you can always un-ban them by going to Tools->Ip Filters) - it may be that the swarm is poisoned by large numbers of seeds that deliberately send no data but use up your connection resources.
Another indication of a poisoned torrent can be when you see a number of peers with similar IP addresses - that is they share a common prefix. Generally swarms will have a random distribution of IP addresses, so if you see 4 peers with an address that starts with 223.103. (for example!) then this is a possible sign of swam poisoning (if your torrent is downloading fast though this can be a sign that someone is running some seed boxes to ensure good speeds and is thus a good thing - employ common sense!)
If your download speeds seems to be pegged at a particular limit it is worth checking that you didn't set some kind of speed limit after that 3rd beer last night - Vuze has global, per-category, per-download (and even per-peer!) limits that can be set - an overview of all limits in effect can be got from the Tools->Speed Limits->View Current... menu (220.127.116.11+).
If you think that after all this the problem is something to do with either your configuration, or network provider, you can always try running a test torrent. A good place to get one of these is from Linux Tracker - if you look at the torrents there sorted by number of seeds (click on the 'S' column) and select one with a decent number of seeds, this will give you a download that should download fast. If this makes no progress then you almost definitely have a local issue that requires further investigation.
Alternatively you could seek out other sources for your download - often the same file will be included in multiple torrents, some of which may well be more active than the one you have happened to download. Since Vuze version 5501 there is a feature called Swarm Merging that allows you to effectively combine multiple downloads that contain the same file(s)