There are two types of assassination missions. Both of them, despite extreme variances in difficulty, pay the same. The names given here are not official - they are both represented on the bulletin board as (faction):Assassination.
Specific assassination missions ask you to kill a single NPC at a specific location.
General assassination missions ask you to kill one or several of a type of NPC, regardless of location.
Difficulty Range Edit
These missions vary in difficulty more than any others. At lower ranks, only missions for weak and common NPCs such as Space Maggots and Young Space Dragons are available, all giving generous time limits of over half an hour. These missions take few APs to complete and can generally be accepted without the fear of not being able to find some to kill. As one's rank grows higher, missions for harder NPCs become available, and soon enough missions for easier ones become unavailable. At high ranks, one might see missions that are practically impossible, such as 20 minutes to kill 2 Z15 Fighters.
It is always best to try to take overlapping missions, i.e.: ones that ask you to kill the same thing or kind of thing. For example, if you have a mission to kill the space maggot at Vega (4,16), and 4 missions to kill 3 Space Maggots, killing the space maggot at Vega (4,16) and 2 other space maggots will complete all 5 missions.
Another universal piece of advice is to fly around high-population planets and starbases and collect the overlapping missions shown above before completing any.
When they become available, assassinations are often well-combined with VIP Transport missions (while flying between planets for assassination missions), and the kills themselves can be used to wholly or partially complete a VIP Action Trip mission. But be careful to remember that once you have taken a VIP mission, you will not be able to take drugs if you run out of APs.
When taking missions for common NPCs, this should not be a problem; however, when missioning for more rare and difficult NPCs such as Blue Crystals, it is a good idea to wait to spot some specific assassination missions in order to be sure that enough of that NPC is alive, and where. It is often also safe to assume that any adjacent NPCs of that same type are also alive, i.e.: the 6 Blue Crystals in Pass Fed-01 are normally killed all at one time, so if one or a few are alive, it is likely they all are alive.