You need to read this article from end to end:
Then you need to understand, via your host or other documentation, what permissions your particular Web server requires to be set so that recommendations in that article can be put into effect within your installation. They will vary from one Web server to the next.
If you go to Google and type in "man chmod" you will find a variety of pretty good explanations of file permissions, at least for unix/linux/*nix. The one at this site http://www.ss64.com/bash/chmod.html
is particularly good if you are on a Linux server because it's got a little practice grid that lets you try out the various permissions. You will learn from this site that you are dealing basically with three sorts of people: User (that's you, the owner of the file), Group (not much used in these circumstances and should probably be set as for Other), and Other (that's all the people, good and bad, who access your site and your blog). There are also, basically, three kinds of permission, and they're pretty self-explanatory: Read, Write, and Execute. Learned users will be quick to correct me, because the whole thing is more complicated than what I've described. But if you can start out by understanding this much, you'll be able to do what you need to do.
If you gain access to your site via a "control panel" or "file manager" of some sort (such as cPanel), it will normally have a little utility similar to the one on that page that lets you evaluate and change the permissions on each file and directory. True aficionados can execute chmod from the command line via ssh or some FTP programs, but that can be daunting if you are just starting out. Contact your hosting provider to see how they suggest you do it.
Potential problems arise with plugins that ask you to set up a directory somewhere where everybody can do everything--that is, User, Group, and Other can Read, Write, and Execute. Or, in the numeric setting-of-bits of chmod, 7-7-7. In a fit of temporary insanity, compounded by amnesia, this is what I did--then uninstalled the plug-in forgetting the vulnerable directory. I might as well have put up a sign saying COME ON IN!
So to sum it all up you have several interesting tasks ahead of you:
1) Read that article to understand how WordPress "wants" its permissions.
2) Understand exactly which permissions are needed on your server to achieve what WordPress is asking for.
3) Understand the three basic kinds of users and three basic permissions.
4) Understand how to actually chmod files on your site, whether via a control panel, ssh, FTP, or whatever.
5) Go ahead and do it--presumably backing-up and testing goes without saying.
6) Remember that it is always, always best to give the least "amount" of permission possible--never give anybody more permission than they need to do whatever task it is you want them to do. In other words, the only permission that isn't "dangerous" is no permission at all.
7) Be dubious of plugins that want to set up places the whole world can write to.
Whew. That should keep you busy for a few minutes. I believe I'll take the rest of the evening off.