Imagine that you met a guy who said he didn't believe in the sun. What would you do? You'd walk him outside, grab a hold of his head, pry open his eyelids, and make him stare right at that big ball of fire until he was blind. Then you'd say, "Why do you think you're blind now, dumbass?"
What if you met a woman who said that she didn't believe in gravity? You'd take her to the tallest building and push her off. Using a bit of pre-planning, you'll make sure to attach an MP3 player on to her and stick the headphones in her ears. As she plummets to her death, the last thing she'd hear is: "What do you think is pulling you down to the ground now, dumbass?"
Maybe you have a friend who doesn't believe in evolution. What do you do but lock them up in the Smithsonian's Natural History Museum and make them memorize all of the transitional fossil and genetic evidence until you finally allow them to leave. (This person gets off much easier, but you can still call him/her a "dumbass" when it's all said and done.)
These, or perhaps somewhat less extreme, examples are what you'd do to prove the existence of what you're talking about. Do you know what you wouldn't do? You wouldn't say: "You can't prove that the sun/gravity/evolution ISN'T real." What kind of argument is that, anyway? It's not necessary when you can prove that they ARE real, is it?
And that's the argument from ignorance. It contributes absolutely nothing to the conversation, and it's not a piece of evidence. The problem is, people will use it to help bolster their claims for everything from God (most popular) to alien visitation, to dowsing. The problem is that it can work easily as well for Santa Claus, werewolves, or the Kardashians, yet nobody would take it seriously if that kind of argument was used. In other words: if an argument is a good one, then it works in all cases, not just the ones that you have already decided are legitimate.
Another form of the argument from ignorance goes along the lines of: "You can't explain X; therefore, I can explain X by saying that it's (insert preferred hypothesis here)." This is another argument that doesn't move a person's case forward at all. There are many things that defy explanation, but the thing is that there are fewer things that we can't explain now than there used to be. When we couldn't explain lightning, all that meant was that we couldn't explain it. It didn't mean that the explanations that people offered up (wrath of an angry sky-god) were somehow more legitimate.
The problem is that it's using ignorance about something as evidence, but ignorance is just that. It proves nothing but a gap in knowledge. For some reason though, people don't like gaps in knowledge, and unfortunately the response isn't necessarily to figure out what the answer is but to supply an answer that can't be justified by any actual evidence.
The two most popular examples of this form of argument from ignorance are the "God of the gaps" and the favorite of the History Channel: "aliens of the gaps". Things are complicated and mysterious, and rather than simply acknowledging that and saying: "We have a lot to learn", they're inserted as "evidence".
I once was accused of being condescending by pointing out that an argument was merely a "God of the gaps". I didn't know how to respond other than to point out that's exactly what had happened. The other person provided no evidence for the existence of God other than to say that a God can explain things that are complicated. Well, if you want to believe that a God is a good explanation, then that's fine, but you need to provide a reason why he's a good explanation - not just the fact that he's AN explanation.
I was then accused of doing the same thing, filling in gaps in knowledge with an "atheism of the gaps". That was a real head-scratcher, and an obvious example of how misunderstood atheism actually is. Atheism provides no explanations. When I can't explain something, I say that I can't explain it. I don't say: "I can't explain it; therefore, it wasn't God." In other words, there is no "therefore" in my point of view. If you want to say that God explains it, then you have to provide evidence for that.
Take my example of the guy who gets blinded by sunlight. The bright light from the sun explains that. And it's not just the "sun of the gaps". We can explain exactly WHY the sun's light does that, and there are other evidences for the sun (like photosynthesis, its warmth, the results are repeatable and verifiable, the fact that you can SEE THE DAMNED THING, etc.) We know what the sun is, what it's comprised of, what it does, and so on. When it comes to God (or aliens) we don't know exactly what he/she/it is or anything about it. When asked questions like that, the other person will appeal to some sort of mystery - which is the heart of the problem. It's replacing a mystery with another mystery, which gets us absolutely no closer to solving the problem.
Keep in mind that using the argument from ignorance does not automatically make you wrong. If a person actually did use "You can't prove that the sun doesn't exist!" as an actual argument, that wouldn't suddenly mean that the sun doesn't exist. Likewise, a person who uses the God of the Gaps hasn't suddenly invalidated the existence of God. God could very well be real - but he's not real for the reason that's being given. In other words, you can make a bad argument for something that's true.
The only thing that I'd point out though is that if the only reasons you have for believing something are all arguments from ignorance, at the very least, you should take a long hard look at why you still believe that.