It can mean several of things, depending upon whether it is being used as a noun or as a verb.
As a verb, the term, "to aspirate", also means a couple of things. You can aspirate a joint, cyst, tumor, etc of the fluid in it. If you aspirate the knee joint (also is called tapping the knee), it means that the fluid collection in the knee joint is taken out using a syringe.
The term, as a verb, can also be used in the setting, where a patient inhales fluid or solids into the lungs. Such as when a patient who is not totally conscious, vomits, then inhales the emesis down into the lungs. It is said that "the patient aspirated the emesis".
So, basically, as a verb, "to aspirate", means to suck, to inhale, to draw up into a container.
As a noun, the term aspirate can be the collection of fluid that was removed from a joint, cyst, tumor, the stomach, etc. The physician may, for example, "send the aspirate to the lab for analysis".