Short answer: It's actually very busy!
Long answer: It's constantly turning parts of itself on and off; on to perform vital functions and off again to save power so that the battery lasts longer.
By far the most important thing it does is to wake periodically and turn on its receiver briefly to see if it has been paged, which means to find out if there is an incoming call. This happens on what is known as a slot cycle, and the period of the slot cycle is controlled by the cell (for all intents and purposes). Slot cycle indices are numbers from 0 to 7, and for any index the period is 1.28 seconds multiplied by 2^index. In North America, by far the most common slot cycle indices are 1 and 2, which indicates a period of 2.56 seconds or 5.12 seconds respectively. I haven't heard of anyone using anything longer than this, though the specification supports slot cycles of 163.84 seconds.
The receiver consumes quite a lot of power. relatively speaking, and the purpose of the slot cycle is to permit the phone to keep the receiver turned off most of the time. This is vital to extend battery life. When the phone first registers with a cell, the cell and phone determine which paging channel the phone will use (if there is more than one) and what phase of the slot cycle that phone will use. Thereafter, the phone wakes periodically, turns its receiver on briefly to see if it has an incoming call or if there is other traffic from the cell it must respond to, and if there is nothing then it shuts the receiver down again and waits until the next slot time.
When an incoming call arrives at the cell for a given phone, the phone system generates the sound of a phone ringing as a comfort tone back to the caller, and the cell waits until the slot time for the phone. When it comes around, the cell sends a message to the phone telling it that there is an incoming call. This causes the phone to waken and set up the call, and to begin to ring.
If the phone doesn't respond to the page, the cell may try again on the next slot.
The advantage of a longer slot cycle is that the phone spends a lower percentage of the time with its receiver on and thus the battery will last longer. It also means there is more capacity on the paging channel. The advantage of a shorter slot cycle is that the phone gets more chances to receive the page, and will receive the page sooner.
When the cell system needs to send out that page, it obviously needs to know where to broadcast it. The cell system as a whole will be divided into zones, and when a phone is paged, every sector of every cell in the zone it's in will carry the page. This means that no matter where the phone is located in that zone, it will receive the page. When the phone moves from one zone to another, it registers again, which permits the cell to know where it is located. The size and layout of the zones is another tradeoff: if the zones are large, the traffic channels will carry a great deal of redundant paging information and can become overloaded, but the phone doesn't have to perform zone-based registration very often as it moves around, which means its battery will last longer. On the other hand, if the zones are small then the paging channels are used more efficiently but the phone will need to register more often and thus will use more battery power.
You may have noticed that when you turn your phone off it takes several seconds for it to actually shut down. That's because it is sending a message to the cell to tell the cell that the phone is going offline. However, the phone can go down unexpectedly without having the chance to send this to the cell (for instance, the battery could be popped from the phone unexpectedly while the phone is operating, which is generally not recommended), and in that case the phone would be offline but the cell wouldn't know it. That would then mean that the cell would try to handle an incoming call for that phone by paging it even though the phone was off, and it generally means that the cell's database would be loaded with entries for phones which aren't available. As a long term recovery for that, the phone is required in most systems to do timer-based registration, which means that every ten or twenty minutes it turns its transmitter on to let the cell know that it's still there. If the cell misses a couple of these registrations in a row, it decides that the phone has gone offline and removes it from the database of "phones which are currently turned on".
Under some circumstances, the cell system can directly challenge the phone for a registration. This happens on the paging channel at the slot, and when the phone receives this message, it turns its transmitter on and sends a registration immediately.
If there is pending voice mail for the phone, the phone will be told on a slot to alert its user of this fact.
All of these regist