What are Newsreaders? Why would you want one? What are newsgroups and Usenet?
Newsreaders are programs that let you read newsgroups, thus I will start by defining them. Newsgroups are forums for discussion that are most comparable to e-mail lists, except that the messages go to the public (unless the group is a local group) and one does not need to be subscribed, in the mailing list sense, to participate.
For example, if there is a group called "rec.widgetmaking" (a group for widget making enthusiasts) that you are interested in, you would tell your newsreader to subscribe you to that group. (You would find the group by looking or searching through the full list of newsgroups on your server, or perhaps a friend also interested in widget making told you about the group.) This has no impact on the group itself, it just tells your newsreader to put that group on your subscribed list. Your newsreader program then makes this group more accessible and stores information about which messages you have already read, etc.
When you open this newsgroup to read, there should be a bunch of messages from various people. (If not, see if your newsreader has a command for loading all messages in a group; some newsreaders seem to default to showing no messages for a group newly subscribed to, as opposed to showing all messages still on your news server). You should have something that looks somewhat like a mailbox in your e-mail program. You can read these messages just like e-mail messages.
The difference is that the message is not addressed to you. It is addressed to "rec.widgetmaking". Similarly, if you choose to post a message in response to a message you are reading, you need to decide (and be careful) whether to reply (or "followup") to the newsgroup, reply to the author (almost all newsreaders have ability to send e-mail messages), or both. (This can be a "netiquette" question. If someone is asking advice, I tend to respond to both the group and the person. But there are times when it is best to take things off the group, and there are some people who prefer not to get responses in their mailbox, since they will see the response in the group anyway.)
That is the main difference between newsgroups and mailing lists. Whereas mailing lists get e-mailed to a certain group of people and the messages exist only in their mailboxes, newsgroup posts get sent to your news server, which then forwards it to news servers around the world (except for newsgroups which only exist on your local server, called local groups).
Usenet is the main source of newsgroups. Just as ABC is a source of television programming, Usenet is a source of newsgroups and newsfeeds. Sometimes people refer to Usenet as newsgroups and vice versa, but strictly speaking, this is incorrect. First, there are other sources of newsgroups that are widely distributed. For example, Clarinet is carried by many ISPs and subscription-based newsfeeds. Second, there are many semi-private newsgroups that are not propogated beyond their own server. You need to point your newsreader to that server rather than the one you normally use; in some cases anyone is allowed to access the newsgroups on that server, but in other cases one needs a password to access the server.
Why would people want to use newsgroups instead of mailing lists? The best answer is to just try newsgroups and see if you like using them. No law that says everyone has to prefer it. But some people prefer not having messages arriving by e-mail all the time. Others prefer being able to drop in on groups when they feel like it, rather than feeling they have to read every message. One nice thing about newsgroups are that if you use a "threaded" newsreader, then the messages show up in logical order; replies to a post show up after that post, usually in a collapsible thread.
Lastly, you should be aware of the difference between "online" and "offline" newsreaders. Online newsreaders assume you maintain an Internet connection during the time the program is running. When you run the newsreader, it polls the server to see how many new messages there are in the groups to which you are subscribed. You can then enter a group, and the newsreader downloads just the message headers, and you see what looks like a mailbox. The messages themselves, however, are not on your machine. As you access a particular message, the newsreader gets it from the server. When you are done reading a message, it is not stored on your machine, unless you save it specifically.
Offline newsreaders, on the other hand, connect to the server, download all new messages for the newsgroups to which you are subscribed, upload any posts you wrote since the last time you connected to the server, and then disconnect. You can then read the messages as you wish, composing replies and new posts to be uploaded next time you choose to connect.
Lastly, a word about expiration. Given that newsgroups are perpetual, how do you wade through all the posts? You don't. The news server "expires" posts after a certain amount of time, determined by the person running the server. Could be 2 days, 7 days, or whatever that person wants. So the first time you enter a newsgroup, or if you use your newsreaders command to access all articles on the server, you are just getting the unexpired posts.