In a basic sense, for many RDBMS:
a) The syntax analysis stage takes input from the server setup (sockets, whatever) and turns this SQL into a valid AST or another intermediate form.
b) It then passes this information to a storage engine which turns this query description into a set of lookups on indexes, tables, partitions, replicated data and other elements that make up the semantics of storing the schema
c) The engine then returns a set of data which is then provided to the client in whatever form (XML, CSV, Client specific).
But there isn't one true answer. You will find similarities in indexing algorithms, distribution algorithms, caching, locking and other things ... but the main similarities is the language interface of the SQL language itself. Beyond there, they can be implemented in any way they wish ... providing their results meet the expected semantics of the input query.
Really RDBMs contain all kinds of structures from computer science ... and each has highly developed and specialized methods for turning the implied semantics of SQL into concrete storage.
Think of how different MySQL and Oracle are ... or PostgreSQL and Microsoft SQL. They all attempt to meet some kind of common SQL-like specification ... but how that specification is fulfilled is diverse.
Engines incorporate all kinds of exotica, specialist indexes to find datas physical location, caching systems and more.
There are tons of open source databases such as MySQL, PostgreSQL and search systems such as Sphinx you can have a look at their implementation. Open source is for learning as much as anything! Try and find a 'mentor' to guide you through the source.