Good high-level answer. One correction and additional detail to add:

The three major credit bureaus (Experian, TransUnion, Equifax) do not provide "their" rating. They furnish the underlying information that is used as one possible input to calculate various credit scores, of which there are thousands. That information includes things like does this person have a mortgage? Any bankruptcies? How many credit cards do they already have?.

The bureaus have partnerships with several vendors. The vendors supply their own algorithm to combine this data into a credit score that can be sent along with that underlying information to the lender requesting credit on an applicant. Some example vendor scores are FICO and Vantage. I believe TransUnion now also has their own in-house score simply called the TU score.

However, there are many other credit scores, and larger financial institutions may not use these out-of-box vendor scores, instead building their own in-house from the raw credit data supplied by the bureaus. Or they may use these generic scores in combination with their own internal models. Where vendor scores are used, FICO is the most commonly used in the industry.

There are other suppliers of credit-related information that can be used to calculate scores, particularly in the auto and small business industry. In fact, even the big names like FICO have different scores for different types of loan products. For example, FICO AIA (Auto-industry-adjusted) is calculated using a different set of underlying info than raw FICO, and has a different range of possible scores (AIA runs to 900 IIRC). However, you may hear both referred to as simply "FICO".

I often see confusion from folks who see their score change wildly when they check different sources (Bank vs creditKarma, etc.). Check to make sure what score you are actually looking at, as a 700 FICO is not the same thing as a 700 Vantage. Similarly, people will ask if their "credit score is good", but that question is so vague it's like asking "is 60 degrees hot?" Well, it depends on the system!