What is GPS
A normal GPS receiver listens to a particular frequency for radio signals. Satellites send time coded messages at this frequency. Each satellite has an atomic clock, and sends the current exact time as well.
The GPS receiver figures out which satellites it can hear, and then starts gathering those messages. The messages include time, current satellite positions, and a few other bits of information. The message stream is slow - this is to save power, and also because all the satellites transmit on the same frequency and they're easier to pick out if they go slow. Because of this, and the amount of information needed to operate well, it can take 30-60 seconds to get a location on a regular GPS.
When it knows the position and time code of at least 3 satellites, a GPS receiver can assume it's on the earth's surface and get a good reading. 4 satellites are needed if you aren't on the ground and you want altitude as well.
What is AGPS (assisted GPS)
As you saw above, it can take a long time to get a position fix with a normal GPS. There are ways to speed this up, but unless you're carrying an atomic clock with you all the time, or leave the GPS on all the time, then there's always going to be a delay of between 5-60 seconds before you get a location.
In order to save cost, most cell phones share the GPS receiver components with the cellular components, and you can't get a fix and talk at the same time. People don't like that (especially when there's an emergency) so the lowest form of GPS does the following:
Get some information from the cell phone company to feed to the GPS receiver - some of this is gross positioning information based on what cellular towers can 'hear' your phone, so by this time they already phone your location to within a city block or so.
Switch from cellular to GPS receiver for 0.1 second (or some small, practically unoticable period of time) and collect the raw GPS data (no processing on the phone). Switch back to the phone mode, and send the raw data to the phone company The phone company processes that data (acts as an offline GPS receiver) and send the location back to your phone.
This saves a lot of money on the phone design, but it has a heavy load on cellular bandwidth, and with a lot of requests coming it requires a lot of fast servers. Still, overall it can be cheaper and faster to implement. They are reluctant, however, to release GPS based features on these phones due to this load - so you won't see turn by turn navigation here.
More recent designs include a full GPS chip. They still get data from the phone company - such as current location based on tower positioning, and current satellite locations - this provides sub 1 second fix times. This information is only needed once, and the GPS can keep track of everything after that with very little power. If the cellular network is unavailable, then they can still get a fix after awhile. If the GPS satellites aren't visible to the receiver, then they can still get a rough fix from the cellular towers.