Note: I'm helping entertain while Jeff is away, so, uh... have a post?
CRACK!
The baseball leaps off Jermaine Dye's bat 10 mph faster than it left Jarrod Washburn's left hand. Dye absolutely killed the pitch - it's a sure home run to left-centre field.
Then the baseball meets the cold breeze wandering into Safeco left-to-right from Puget Sound, drops like Jose Lopez attempting to slide into second base, and finds itself nestled in Ichiro's glove. Two outs, two runners stranded. Washburn and the Ms are saved by Safeco Field once again.
Every baseball fan over the age of 10 is at least dimly aware of the fact that different stadia can have a significant effect on the way the game is played. Coors Field in the early 2000s played extraordinarily well for hitters (less so now, but it's still an extreme hitters' park). Petco Park is death on hitters (especially Khalil Greene), and Safeco Field isn't exactly the most friendly place to bat either. We take offensive performance in High Desert (our A+ affiliate) with a grain of salt, and we know that Tacoma's pitchers are helped out by making their home in Cheney.
The differences caused by ballpark vary from the fairly mundane (Petco suppressing runs) to the strange (batters in Safeco hit more fly balls than expected) to the downright weird (RFK Stadium decreases the run expectancy of line drives). We don't have logical explanations for some of these differences, but we don't need to in order to measure them (the details of which would be a whole different post). Mostly, we aren't terribly concerned with the fine details, because what we're after (generally) is a representation of the run environment that a specific stadium creates compared to its league. There are essentially two uses for park factors as applied to individual players: assessing how valuable he is in a given park and attempting to determine how he would perform in another. These are two very different questions (and mixing them up is a great way of getting yourself confused). Let's address the first.
A player who creates x runs is more valuable in a lower run environment than a higher one. This much should be fairly obvious - less runs are being scored and therefore the impact of those x runs is comparitively bigger (it works like this with pitchers as well, obviously, except backwards). The important point to take away from this is that it doesn't matter whether the player in question is actually affected by whatever's causing the change in run environment. Raul Ibanez value is higher due to the low run environment he plays in despite Safeco actually helping left-handed pull hitters. You'll see far too many arguments saying park effects shouldn't apply to a player because [insert something about home-road splits/something similar here]. These sorts of arguments are completely meaningless - the value a player provides has nothing to do with whether he's affected by his park or not and everything to do with the context in which he plays. This really is the key idea to take away when thinking about park effects: When measuring a player's value, everyone should be subjected to a blanket park effect, because you're just looking at their comparative contribution in a certain environment.
Ok, so that question answered. On to the next - projection. Remember when I said we weren't terribly concerned with the fine details? Well, that's not entirely true...
The really basic example to consider here is why our front office is so obessed with finding left handed sock (note to FO: Teixeira and Griffey for '09 please). In essence, our park doesn't play the same for lefties and righties. Specifically, it's much, much easier to hit a home run to right field than it is to left or left centre. This means that any left handed pull hitter should be expected to do much better in Safeco than a right handed clone of said player would - unsurprising, considering who the park was built to accomodate. This also means that lefty flyball pitchers (i.e. Jarrod Washburn) will find life much easier in Seattle, since most of the home runs hit against them will be into left field. So when you're trying to predict how a player would perform in a different park, you would have to park neutralise their stats (i.e. correct for every stat affected by their home park) and then apply a correction for their new park. This sort of analysis is pretty time consuming, especially it you're attempting to go about it by hand, but it illustrates exactly why it's of paramount importance to have accurate park effects (for LHBs and RHBs both) for as many stats as you possibly can. Which we don't, as far as I'm aware. If someone wants to get on that, that'd be pretty awesome...
There are many, many reasons why parks play differently to one another - shape, altitude, humidity, prevailing wind, and the ever-popular 'we don't have a goddamn clue'. For the most part, however, we can condense all of this information into one number (essentially representing runs per game/league average R/G) which can be found in the teams section of Baseball-Reference. Armed with that number, you can do a good bit of meaningful analysis on player value (and, in leiu of more advanced data, make a decent stab at predictions too).
Parks matter, often far more than we give them credit for. Ignore their effects at your own peril.