What is a Moderator in an Internet Forum?
A moderator (or "mod") is someone with powers to moderate the discussion in an online forum, which means that they can remove posts that violate rules, lock threads so that no new posts can be made, and temporarily or permanently suspend (ban) users from participating in the forum.
Tiers
In general, moderators have tiers, since there must be someone with power to moderate the moderators in case they abuse their powers, e.g. a "super" moderator, or an "administrator," or the owner of the forum. This could be owner of the forum website, for example.
Traditionally, moderators would be normal users of a forum who apply to a moderator position by contacting the owner or existing moderators somehow and are promoted from normal users to moderators with additional powers. In commercial forums, moderators may be staff who are hired and paid to moderate the forum, but unpaid moderators who are just users of a forum and never went through a job interview are also very common.
Although they may be unpaid, the moderators are still "rewarded" by having power over other users in a forum. Since you can get banned by disrespecting a moderator, users that want to remain in the forum will typically interact with moderators in a different tone than they would with normal users.
Nowadays, many online services allow users to have their own forums where they have (almost) total power, e.g. subreddits on Reddit, and "servers" on Discord. The person who creates the subreddit or "server" becomes its owner and can set any rules they want, and promote other users to moderators, however, they are one tier below the staff of the online service. The owner of Reddit has authority over the owner of a subreddit who is just a random Reddit user. Naturally, the owner of Reddit won't go around moderating subreddits personally, so there is staff below the owner but above the moderator users. This staff that moderates users are called "administrators" or "admins" on Reddit. A similar concept exists in any such service.
Distinguishment
Generally, moderators and staff are distinguished from normal users in some way, e.g. by having different colors in their names or special icons. In some cases, this distinguishment only appears when the moderator wants it to appear, e.g. when speaking in official capacity as opposed to just being another user on the platform.