First off, not everyone can or should be a beta reader. Authors rely on their betas to give them serious feedback, both good and bad. A beta must have a strong command of spelling, grammar, and punctuation. No exceptions. A good beta will also understand pacing, plotting, characterization, have a thorough knowledge of the canon material. And a truly awesome beta will know how to do research to find out things like historical inaccuracies and to settle grammar and punctuation questions. That's truly going above and beyond, and if you find a truly awesome beta, never let her go.
But anyway. If you find yourself questioning your ability when it comes to the basics of writing, don't volunteer to beta read. Not being a beta is no shame; but if you don't know the basics, you may harm a fic more than you help it. Only you can judge if you've got the skills needed, and many of those skills can be learned with experience and practice.
You should also make sure that, when you agree to take on a beta, you have the time and motivation to get it done. Let the author know how long you expect it to take, and keep in contact with the author, especially if you think you'll take longer than expected. This might sound obvious, but it turns out to be harder than you might think. A lot of people complain that betas take too long or disappear; I've both had that complaint and been the absentee beta. So just think about it before you begin.
There's no definitive way to beta a story. I can only tell you what works for me when I beta read, and what I've found helpful when other people have beta read for me.
When you first get the fic, contact the author. If it wasn't included, see if she has a list of questions you can answer. This should be things she's unsure about in the fic: how's the pacing? does a specific scene work well? does a specific character seem OOC in any scene? Things like this are key to keep in mind, so you can make sure the author gets everything she needs out of your beta work.
The next thing I do is read through the fic as it's been sent. I usually don't make any changes the first time through, I just read it to get an idea of what's going on, what might be potential problem areas, and to find your general reaction.
Once you know what's going on, it's time to make corrections. You want your author to know what you changed and why, so you'll want to track changes. If you're using Microsoft Word, you can do this easily by going to tools → track changes → highlight changes. From there, you get a small menu with a series of boxes; make sure you check track changes on screen and track changes while editing. From then on, everything you type will show up in a non-black color, underlined, with a little line at the margin to show that a change was made there. If you don't use Word, or you can't find the menu, most programs have an option to let you change text colors. Every time you make a change, make sure you color your changed text so the author can change. Or, if all else fails, bold all the changes you make. Just do something so the author can see what you're doing.
(Tip: you might want to rename the document so it's clear you're working on a beta version, and also so that you're sure you have a backup copy of the original version. I usually name it something like title_beta.doc.)
Now you're ready to read through the story and fix things! This is the major part of beta reading. As you go through, fix errors. Hopefully the spelling has already been checked, but keep an eye out for misspelled words, or incorrect words. Fix all punctuation problems you encounter, and correct any grammar mistakes.
Trickier are things that are stylistic, plot, or character questions. If you look at a sentence and find it awkward, you might want to make a note. If you change the author's wording, make sure you explain why. There's a fine line between fixing errors so the story reads more smoothly, and canceling out the author's style and intentions. You want to do the former, not the latter. It'll help the author accept changes you make to her masterpiece if you explain why you do them; and it'll help make sure you're changing things for the right reasons, not just your own preferences.
When it comes to scenes that you feel are problematic, areas where you're not sure what's going on, and characters who seem to be out of character, you might want to make notes at the scene's conclusion. Explain what seemed awkward or wrong to you, and brainstorm suggestions at how to fix it.
Once you've got the fic read through, the obvious mistakes changed, and suggestions for the stylistic problems, think back. Were there any trends? Most authors have things they aren't good at. For example, I'm terrible with consistent POV. A couple people I've beta read for had problems with punctuating dialogue. Many, many people have problems with commas. If you've noticed a problem coming up in the fic several times, you might want to make a note at the end explaining what the problem was to the author, and how to fix it.
Now here's another key. You want to encourage the author, while making the work better, so don't forget to praise! Just like when you give feedback when you're reviewing a fic, make sure you notice what the author has done well. You can put positive comments into the fic itself, just like when you were writing comments about things that didn't work, or (and/or, in fact) you can leave them at the end. Just make sure that you let the author know what works well as well as what doesn't, so she's got an idea what her strengths are, and which scenes to model the problem areas after.
And that's a beta. Be prepared for questions from the author, and to maybe do another read through if she's still unsure, but there you go.