In professional wrestling, a manager is a secondary character paired with a wrestler (or wrestlers) for a variety of reasons. The manager is often either a non-wrestler, an older wrestler who has retired or is nearing retirement or, in some cases, a new wrestler who is breaking into the business (or a specific company) and needs the experience in front of the crowds.
The role of a manager
A manager is somewhat like a (kayfabe) agent for an actor or an athlete; he helps his client to book matches and appearances, and otherwise further and guide their career. As parts of storylines it's the manager who positions their charge for title opportunities, decides who to trust as an ally, and generally acts as a mouth piece on their wrestlers behalf. Outside of storylines, a manager's job is to help the wrestler they're paired with get over. Often the very act of aligning oneself with a manager, or conversely breaking away from a manager they've worked with, can change a wrestlers alignment, making them a sudden face or heel.
While the basic goal of a manager never changes (get their wrestler over), how they go about it will on alignment. A heel manager, for example, may have their wrestler constantly duck tougher opponents while cheating to help them win when they do actually wrestle. A face manager, however, may spend the majority of their interview time talking about how tough their wrestler is and going out of their way to find bigger and better opponents to challenge to prove it.
Further still is a special kind of manager, the "handler". If a wrestler's character calls for it a handler is the only person who can control a "wild" wrestler that is prone to "go out of control".
Managers vs. Valets vs. Enforcers
Most frequently managers are male, with a female who accompanies a wrestler to the ring being referred to as a valet. The major difference between the two being that the manager is usually shown as being someone who is (kayfabe) crafting a wrestlers career "behind the scenes", while valets are usually shown as simply eye candy, often to distract opponents or Referees. Valets, in general, don't cut promos, instead simply making their presence felt by standing around while their wrestler does. Although female managers are rare, some (like "Sensational" Sherri) have had long and storied careers and gone through many companies as "managers of champions".
Further still there are Enforcers, full-time wrestlers who accompany another wrestler to the ring. This is generally done when a wrestler needs to "have [their] back watched", as enforcers will keep outside interference from taking place in a match, or become outside interference themselves. They're generally larger, more intimidating looking wrestlers.
In the early years of pro wrestling, up until the mid 1990s, managers were very common because they served a secondary purpose to "getting heat". A manager's job was to cut promos on behalf of clients who may not be as charismatic (or had gimmicks in which they were either foreign and didn't speak English or were silent). As such, all managers were required to have the "gift of the gab", though it wasn't unheard of to see managers paired with wrestlers who had no trouble with cutting their own promos. In this case, the manager's role was often mainly to help their client cheat in matches.
Some managers, such as Paul Ellering, were managers in the truest sense of the word, taking care of the day-to-day needs of their "charges" in terms of travel arrangements, rentals, and lodgings.