A fund manager is the person, or team of people, responsible for actually running a mutual fund on a day-to-day basis — deciding which stocks or bonds to buy, how much to hold, when to sell, and how to position the fund based on market conditions and the fund's stated investment objective and category.
In an actively managed fund, the fund manager's skill and judgement directly influence returns, which is why their track record and consistency across market cycles matter when evaluating such a fund. In a passive fund, like an index fund, the fund manager's role is much more limited, since the fund simply follows a fixed index rather than making independent stock-picking calls, which is also why passive fund fees tend to be lower.
Some funds are run by a single named manager, while others use a team-based approach, which can make performance less dependent on any one individual leaving the fund house.