Basically there are two reasons you will be put on a heading, tactically (i.e. positioning in a stream or for final) or for separation, sometimes both.
You will be well aware that you are being tactically positioned whilst inbound but in the en-route environment we will be positioning you outbound as well. It is common practice to vector an aircraft climbing through FL200 against traffic cruising at FL370 to get both aircraft line astern at FL370.
The lower aircraft may need to on a heading for lateral separation against aircraft above to facilitate the climb through. If there is more than one aircraft to get through this may result in several changes of heading. Remember that all the aircraft involved must be on a heading, the one climbing and all the aircraft it is climbing through.
When you are put on a heading for separation purposes the heading will be imposed well away from the collision point so it will rarely be immediately obvious in the cockpit what the traffic is. You will often be asked to turn 5 or 10 degrees as a result of a confliction 60 or 70 miles away with the traffic well over 100 miles from your position. If you were given a turn and the traffic was obvious on TCAS the turn instruction would probably be accompanied by “Avoiding Action” delivered with an appropriate degree of urgency!
All this vectoring is occurring within the confines of controlled airspace and these boundaries often mean more vectoring to make use of the limited amount of space available. For instance, you may be vectored into a hole in the traffic to allow climb (or descent) but then need to be vectored back to a position that will allow you to return to your flight planned route without leaving the airway system or penetrating an active danger area.
This is a pretty general overview. If you are really interested come and visit us at Swanwick, you would be more than welcome. With TC and AC in the same building you will be able to see how you are handled from leaving the Tower freq to leaving the London FIR.