In the highest level you can ask, what a database really is. Is it every form of accumulated data? Most people agree that a database is a sortiment of data that is organized or structured in some way.
You can differentiate between data lakes, data vaults and data warehouses. Data lakes are an inhomogeneous group of data which is stored in one or more databases. The purpose is to store a great amount of information quickly. However, the data is not pre-structured. Therefore, searching or analyzing might take longer. Data vaults consist of multiple databases or database schemas with every one of them containing data of a certain value, purpose or type. This takes a lot of initial work when setting up the vault but is highly effective, once it comes to analyzing data. It can also process a lot of information parallely and is often used with cloud computing. If you want to access data quickly based on certain tags, filters or topics, then this is the way to go. Finally, data warehouses are made up of many databases and can be described as a super database, which is why it is often seen as the king's discipline of database management.
The databases themselves can mainly be categorized into relational/non-relational or sequential/non-sequential.
Relational databases follow the goal that every table or entity should/can in some way be linked or connected to any other table or entity. This makes it easy to see the relations or dependencies between various entries. Also, searching, filtering or debugging data becomes easier. Nevertheless, it takes a lot of effort to keep track of all the relations and often, database administrators or developers have a hard time considering all the combinations and links when editing code or documents. Furthermore, relational databases use complexe data base management systems (DBMS) which contain some heavy algebra. Relational databases are e.g. Oracle, PostgreSQL, MySQL. They all depend on the Structured Query Language (SQL). With small differences, they all use the same basic commands to alter, edit, search or write data. There are further sub-categories like type-relational, object-relational, etc. but the differences are rather insignificant.
Non-relational databases are less complexe, easier to maintain and they are not as sensitive to logical or mathematical errors as relational ones. But they can be less useful for big amounts of data or for purposes like data mining, quick search or personal information storage. Data is mostly stored in form of different data types. Instead of strictly sticking to the table-row concept, they can contain users, bookings, documents of various shapes and forms. The greatest weakness of those databases is their lack of "smart connection". Because of non-existant links between documents, specific queries or search efforts can take a long time. Also, double entries, missing entries or mistakes are less likely to be detected immediately by the software system. Non-relational databases can be sub-categorized into key-value-pair types, broad row tables, document stores, search engine bases or graph/image data bases. Examples include Neo4J, Datastax Enterprise Graph, some NoSQL bases like Couchbase and MongoDB or Scyalla and Cassandra. As you might guess, they don't use SQL but NoSQL. You get data in easily and quickly but out slowly and sometimes with complications.
So, to specifically answer your question, relational and non-relational are the two (only) big and official types (by big I mean serious mathematical differences in data processing). Thereby, SQL and NoSQL are the biggest query languages with vast differences. Document, graph, etc. are just forms of data structures that are often associated with NoSQL databases but they are not a seperate type of language or base! In the same way, the forms of databases (e.g. symmetrical, snowflake, tree, star, etc.) are only a way to describe their basic hierarchy or structure. They, too, don't form their own categories...
Dataframes, data lakes and data vaults (ultimately data warehouses) consist of many databases and can be relational, non-relational or a mix of both!
I want to make clear, that it comes down to relational and non-relational. Especially with databases, I hear a lot of nonsense and people who differ between details, they mix up forms, shapes, languages, data base names and whatnot. Document, MongoDB or snowflake are neither languages nor mathematical models.
PS: I add some links in case you want to learn more.
https://www.oracle.com/database/what-is-a-relational-database/
https://www.pluralsight.com/blog/software-development/relational-vs-non-relational-databases
https://www.oracle.com/database/what-is-database.html
https://www.guru99.com/data-warehousing.html