If you don't care about beer, skip to the post below mine!
Gravity indicates the weight/density of your liquid against water. So, 1.000 is the same as water. Generally, a beer with a low FG (final gravity, once it's done fermenting) will have thin body and dry flavour (because of lack of sugar) and a beer with a high FG will have a fuller body/feel and be sweeter. Low FG is good for beers like saisons, and high FG good for stouts.
Any time you take a reading, it's an SG (specific gravity) reading. When you take the first reading (before you add yeast), it's also the OG (original gravity). Once the fermentation is finished you subtract the FG from the OG to determine the ABV (alcohol by volume). There is software you can use to help with these calculations (I use 'Brewzor Calculator' on Android). Remember when taking a reading to also take the temperature, as that will make a difference in the reading.
It's important because you want to know when fermentation has finished. If you don't care what your ABV is, you *could* go without taking gravity readings, but it'd be tough to know for certain when a beer has finished fermenting.
When you have the same SG for at least three days, the yeast is finished (either because it gave up/stalled, or because it's eaten all it can) and it's time to think about bottling.
Sugar and honey are added, and for different reasons. Belgian brewers will add 'candi sugar' (inverted sugar, basically, it's not hard to make yourself) to bump up the alcohol in a beer and to dry out the flavour* without changing the basic flavour a heap. Table sugar isn't often added, honey is sometimes but again it's not common.
* Because table sugar/candi sugar are simple sugars (I can go much longer on this but let's keep it high level) the yeast will chomp through it really thoroughly meaning the remaining residual sweetness is lessened. The sugars created from mashing the grain (essentially steeping it in hot water to release the sugars) are more complex, thus the yeast won't convert it so completely to ethanol. And going further, the temperature at which you mash the grains also has an effect on the complex sugars created. Simple rule - the cooler the mash the drier and thinner the beer. General temperatures are 62c - 69c.