I am really interested in how MySQL indexes work, more specifically, how can they return the data requested without scanning the entire table?
It's off-topic, I know, but if there is someone who could explain this to me in detail, I would be very, very thankful.
In MySQL InnoDB, there are two types of index.
Primary key which is called clustered index. Index key words are stored with real record data in the B+Tree leaf node.
Secondary key which is non clustered index. These index only store primary key's key words along with their own index key words in the B+Tree leaf node. So when searching from secondary index, it will first find its primary key index key words and scan the primary key B+Tree to find the real data records. This will make secondary index slower compared to primary index search. However, if the select columns are all in the secondary index, then no need to look up primary index B+Tree again. This is called covering index.
Adding some visual representation to the list of answers. 
MySQL uses an extra layer of indirection: secondary index records point to primary index records, and the primary index itself holds the on-disk row locations. If a row offset changes, only the primary index needs to be updated.
Caveat: Disk data structure looks flat in the diagram but actually is a B+ tree.
Source: link
I want to add my 2 cents. I am far from being a database expert, but I've recently read up a bit on this topic; enough for me to try and give an ELI5. So, here's may layman's explanation.
I understand it as such that an index is like a mini-mirror of your table, pretty much like an associative array. If you feed it with a matching key then you can just jump to that row in one "command".
But if you didn't have that index / array, the query interpreter must use a for-loop to go through all rows and check for a match (the full-table scan).
Having an index has the "downside" of extra storage (for that mini-mirror), in exchange for the "upside" of looking up content faster.
Note that (in dependence of your db engine) creating primary, foreign or unique keys automatically sets up a respective index as well. That same principle is basically why and how those keys work.
Let's suppose you have a book, probably a novel, a thick one with lots of things to read, hence lots of words. Now, hypothetically, you brought two dictionaries, consisting of only words that are only used, at least one time in the novel. All words in that two dictionaries are stored in typical alphabetical order. In hypothetical dictionary A, words are printed only once while in hypothetical dictionary B words are printed as many numbers of times it is printed in the novel. Remember, words are sorted alphabetically in both the dictionaries. Now you got stuck at some point while reading a novel and need to find the meaning of that word from anyone of those hypothetical dictionaries. What you will do? Surely you will jump to that word in a few steps to find its meaning, rather look for the meaning of each of the words in the novel, from starting, until you reach that bugging word.
This is how the index works in SQL. Consider Dictionary A as PRIMARY INDEX, Dictionary B as KEY/SECONDARY INDEX, and your desire to get for the meaning of the word as a QUERY/SELECT STATEMENT. The index will help to fetch the data at a very fast rate. Without an index, you will have to look for the data from the starting, unnecessarily time-consuming costly task.
For more about indexes and types, look this.
Indexes are used to find rows with specific column values quickly. Without an index, MySQL must begin with the first row and then read through the entire table to find the relevant rows. The larger the table, the more this costs. If the table has an index for the columns in question, MySQL can quickly determine the position to seek to in the middle of the data file without having to look at all the data. This is much faster than reading every row sequentially.
Indexing adds a data structure with columns for the search conditions and a pointer
The pointer is the address on the memory disk of the row with the
rest of the information
The index data structure is sorted to optimize query efficiency
The query looks for the specific row in the index; the index refers to the pointer which will find the rest of the information.
The index reduces the number of rows the query has to search through from 17 to 4.
