Every registered user can rate any game in the BGG database. Although these ratings are entirely subjective, here are the suggested guidelines:

To do this, find the game and go to that game's Main Page ; this is most easily done using the Search option near the top of every page. In the topmost section is a series of stars labeled "My Rating", which you can click on to set your rating on a scale from 1 to 10 (once set, you can click on edit to use decimals). You can alternatively go to your In collection drop-down arrow, click on Edit and fill your rating there.

Sometime around 2017 it was decided not to display on a game's main page its average User Rating if the game has fewer than 30 ratings (since a small number of user ratings can be so extremely variable).

If a user has and rates several copies of a game in their collection , then only their highest rating for the game is used .

Common proposals and complaints

Various ratings-related proposals and complaints appear repeatedly in the forums.

Very many diverse proposals for "improved" formulas, sometimes including purported "mathematical proofs" that the new proposal is clearly superior.

Complaints about low ratings perceived as malicious or trolling, with various proposals to disallow such ratings (which are complicated once one tries to formalize them in detail, e.g. how to define and prove a "bad faith" rating, how to police them, etc).

Complaints about ratings wars between 10-raters ("Cool minis, can't wait for this game!") and 1-raters ("Countering the hype!")

Complaints about ratings for unreleased games, with various proposals to disallow rating unreleased games (which are complicated when trying to formalize them in detail, e.g. how to define an unreleased game, how to police this, what about playtesters and others who really have played it, what about games with multiple versions & editions, etc).

Proposals for different text descriptions / meanings of the values 1-10.

Proposals to only allow people to rate games which they've played, or which they own. (But many users have no interest in logging plays or marking their owned games. And nothing prevents users from falsely logging fictitious plays or falsely marking a game owned.)

Proposals to only allow people to rate games if they also write a user comment about the game. (Which would probably cause an increase of hastily typed useless vapid user comments.)

Current BGG policy is that users are allowed to rate games however they wish, as long as each person only rates a given game once.

Ranks

The User Ratings are also used to determine the Rank of a game (not expansion) in the BGG database. Only games that have at least 30 User Ratings are eligible for Ranking and to the User Ratings are added a number of "dummy" ratings, which are then used to produce a new average Geek Rating. (E.g. see this thread.) This is the rating that shows up in BGG searches and the number can, and does, vary from the Average Rating. In effect the "dummy" ratings move a game's average towards the norm of all games on the database - making games with few votes but very high ratings lower ranked than games with many more ratings but a lower Average Rating. (If you want to know more about this process, search on "Bayesian" within BGG.) Additionally, secret undocumented stuff is done to try to filter out obviously bogus "shill" or "hate" ratings. (There are many threads from people asking about or trying to figure out the details, but they are intentionally undocumented.)

Note that the number of "dummy" ratings apparently depends on the total number of ratings. This explains the apparent "paradox" (often asked in the forums, e.g. here) of why game X is higher ranked than game Y overall, but game Y is higher ranked than game X in subdomain Z.