A remake of a movie is pretty much exactly that: an old movie, usually with dated special FX, gets remade for a contemporary audience (which really likes its eyecandy, I won't deny or decry that). I'm sure you've encountered many remakes about which the critics decry how the film has lost its heart amongst shiny new CGI lasers and such. This isn't high-brow talk, this is a real problem.
An example of the over-reliance on FX as opposed to content would be a comparison of Star Trek series. The phaser beams, not to mention the fight choreography haha, haven't aged well on the older Star Trek series but the plot lines you found on The Next Generation remain vastly superior to any thing that Enterprise had to offer. By the way, that was an example of an old thing with good story but bad effects compared to a new thing with good effects but bad story. That was NOT an example of a remake. Fortunately, Enterprise got such low ratings the network cancelled it. I take much comfort in the knowledge that the Trekkie fandom (most of it) retained good taste.
Sadly, movies are more or less one-shots. Even if the film turns out crappy, the viewers will have already paid for their tickets. More often than not they have been suckered in by spectacular trailers, which the companies are never shy about stuffing with the best snippets of FX (and sex). And therein lies the heart of the problem, which should honestly NOT be a problem, as such. The old film's story was great, that's why it got someone's attention. The FX, perhaps, not so much. Ok. Spruce up the effects and retain the golden story, right? That second part has given generations of movie makers, be they writers of the new screenplay or movie executives, much trouble. They don't seem to understand. They just don't understand.
In contrast to a 'remake', a 'reboot' involves taking only the essence of the original and altering it to fit the tastes of the contemporary audience. Granted, remakes often do a little of that altering too, but no-where near this scale. The best example of a reboot in recent memory would be the series of Batman films: Batman Begins  and The Dark Knight . Notice how both these films do not reuse the plots of any of the older Batman films but more importantly, note that the tone and feel of both of these films can be described as "darker, grittier and more realistic". The young'uns out there probably won't have seen any of the campy Batman episodes from way back in the day but they'll get the idea from this.
That wasn't one joke episode. The Batman of the 60s was like that kids.
Need another example of a reboot? The two recent James Bond films: Casino Royale  and Quantum of Solace  are excellent examples. But wait! There was a film called Casino Royale that came out in 1967. Does this mean Casino Royale  is a remake of that film? No. In fact, the 1967 film is a satire of James Bond films and did not actually feature James Bond.
Again, "darker and more realistic" are the buzzwords of the day. Up until Casino Royale, Bond films had basically become self-parodies. Bond arrives in a suit, beds woman, saves day, all the while snapping off witty quips and one-liners. Then came The Bourne Identity  with the (of course) juiced up fight choreography and doing away with elaborate doomsday MacGuffins. (Wikipedia is your friend if you don't know what that term means. Try it.) I hope the difference between the older Bond films and these new ones is evident to you viewers.
Still don't get it? Ok.
A remake is a film remade.
A reboot is a film reinvented.
Final push: a visual aid with the Batmobile.
|Remake - note the retention of 'bat-ness'|
|Reboot - All business. They're going for 'realistic'.|
In response to some of the comments from the ill-fated post:
@Natural One: You are right about that, but not this time! Rise of the Planet of the Apes is NOT a remake and therefore we should applaud them.
@Rob: I know exactly what you mean, but with regards to the apes, I think the CGI and Andy Serkis are a great improvement over the stiff masks they had in the original. A limitation of the technology of the time.
@ Mai Yang: No, don't do that! Go check it out! It looks like it could be really good.
@Clueless Dolphin: I disagree but with the special effects so good in this one, I guess that's your liberty.
@Timothy Bowen: (O_O)-b
@Tony Van Helsing: 50%. Yes, it does look like it might be pretty good. No, it isn't a remake.