Messier objects are celestial bodies that were observed by Charles Messier throughout his career. During his lifetime, any person who found a new comet became well known amongst their peers, but, also became a celebrity. Messier lived in what could be considered precarious times for professional astronomers. There were few jobs, so if you did make new discoveries you did not have a job.
Each Messier object is a body that first appeared to be a comet, but is not. Messier compiled a list of these objects in several additions; eventually ending with a total of 103 objects during his lifetime. His work was limited by the fact that he lived in the Northern Hemisphere, so could only observe objects that appeared in the night sky above 35.7° latitude.
It can be difficult to observe the entire Messier list. In addition to the 103 Messier compiled, his assistant and other researchers followed up on his side notes and astronomers now believe his list should contain a total of 110 objects. These objects are an interesting challenge for amateur astronomers to find, so there are several astronomical associations that offer rewards to anyone who observes them. A simple web search will garner you the information you need to participate.
Below is a set of links to an introduction to Messier objects as well as a link to an article about each of the individual Messier objects. Enjoy your research, then enjoy your observations.