The Messier Objects
During the 18th century, Charles Joseph Messier, a famous astronomer, created a list of Deep Space Objects. During this period of time, telescopes and stargazing equipment were obviously not very advanced. That is why the Messier objects, or M objects (such as M31, M42, etc.) are great objects for beginning astronomers of any level of interest to observe. No matter what quality your equipment, most Messier objects will be visible in relatively dark skies. The Messier Objects are also the basis of any amateur astronomer's observational goals. Many astronomers will make a checklist of the Messier objects they have seen and have not seen. Sometimes, astronomers will start a "Messier Marathon". This is a personal challenge where an astronomer will try to observe as many Messier objects in a given amount of time as they possibly can. Really, seeing all the Messier objects is the only goal of many astronomers. Because really, there aren't many Non-Messier objects that are interesting! Don't get me wrong, there are some, but I'll discuss those later.
M31
Picture from AllAboutAstro.com
M27
Picture from AllAboutAstro.com
NGC Objects
There are also some interesting objects called the NGC objects. The New General Catalog is a list of over 13,000 Deep Space Objects created in 1888. Besides the Messier list, this is the second most popular listing of Deep Space Objects. Although most of these will be hard to observe by a beginning amateur astronomer, many of them are very interesting. These objects are marked as NGC then a number, such as NGC 153.
NGC 3370 
Picture from HubbleSite.org
NGC 7293
Picture from HubbleSite.org