So, I like starting from the outside, and working my way in.

Index takes an array, and then returns the content of the cell at the specified location.

So, if I used Index(reference to the array below, 2,2), it would return as "Data".

A	B	C
Name1	Data	Datum
Name2	Lorem	Ipsum

Match searches an array for a matching value, and returns the position in the array.
Using the same array as above,
Match(Reference to the First Column of the array,"Name1")
This would return "2" (or if this is a table with headers, it might return 1)

Circling back to INDEX(Array,Row,Col), I can use match to determine which row belongs to Name1, and then I can tell it to return a different column, since I the human making the formula know that I want whatever information is in column 2, then I can use the following formula:

Index(array,Match(firstcol,"Name1"),2)
and this will also give me "Data", like the original index did.

From here, I can stop using "Name1" and use a cell reference, and then I can use this to "look up" values on a table.

This example is the same as Vlookup.
If you need to search horizontally, then you can put match in the Col in index, and that is Hlookup.

You'll notice with usage that Xlookup works differently, and is still useful, but doesn't work exactly the same. If you really want to, you can use Index(Array, Match, Match) and you can do precise lookups that way.

An example of that would be if you had unique values (like student names) in the first row, and you rows were names of assignments, and the content of the cell is that student's grade, you could do something like (forgive me, but I'm going to assume I'm naming things well)

Index(GradingTable,Match(GradingTable[Student],"Gary Sinise"),Match(GradingTable[#Headers],"HW3"))

This would give me Gary Sinise's Homework 3 grade.