A trajectory is essentially a route or pathway of an object over the course of time. For mobile lidar mapping it is absolutely essential and includes positioning (XYZ) and orientation. It can be directly determined or indirectly. It is generally needed at 100 to 200Hz at a minimum. Most people are familiar with it in terms of navigation via their smart phone or other similar device.
Most often the trajectory is a direct result of an Inertial Navigation System (INS). An INS combines an IMU (inertial measurement unit) along with a GNSS (global navigation satellite system). An INS provides the most widely usable and affordable solution to determine a trajectory. GNSS provides global coverage and virtually always uses GPS (provided by the US government). More governments are adding more satellites now such as GLONASS (Russia), Beidou (China), Gallileo (Europe), etc. See here for more information as it becomes available. The GNSS provides positional geo-referencing (i.e. coordinates).
As part of the INS, the IMU provides the orientation or attitude information. An IMU is a passive, non-emitting system working from gravity. There are a large number of vendors that solely provide INS packages. Some are very small (the size of a finger nail) and some are quite large (the size of a mailbox or even bigger). Prices vary widely and so do results.
Positioning accuracy can vary from a few centimeters to several meters depending upon the GNSS solution used. Orientation can vary in a similar manner.
In any case, a trajectory is essentially the spine holding the entire framework of a mobile lidar system together. It is absolutely essential that the quality of the trajectory be appropriately determined for the product to be generated. If high accuracy is required in a final product, then the appropriate hardware and software must be used to reach that goal.