The so-called God's Algorithm would solve any position in
a minimal number of moves. God has not yet revealed it.
There are procedures for solving the cube,
they take about two pages of instructions to describe.
The basic idea is this:
there are various known combinations of moves which
will rearrange certain "cubies" (as few as 2 or 3) and
leave everything else in place. In the course of doing
them, you might mess other cubies up, but then you
put them back, rearranging only the ones you wanted to.
You build up a repertoire of these (about a dozen or two
dozen), and apply them as needed.
A fairly simple, but not too quick method is as follows:
First pick one face and arrange all the 8 cubies that form
its edge properly. This can be done without much in the
way of special moves, since it's ok to mess up everything
Then solve the middle layer, using your bag of tricks.
On this stage you want to preserve the first face,
but it's ok to mess up the last face.
Solve the last face, again using pre-figured out moves.
This is the most constrained because you don't have
much freedom with regard to the rest of the cube.
The larger your bag of tricks, the more efficient you can
be by doing larger combinations at once.
For example if you have 5 cubies to cycle into each
others' places, and you have a 15 move sequence that
does that, it's quicker than two 12 move sequences that
do three of them and then the other two. The ultimate
version of that would the God's Algorithm.
When I was in practice, I could do it in about 3 minutes.
Frankly, I don't have the slightest idea how the people
who do it in 20 seconds or whatever manage that,
even with specially lubricated, easy to turn cubes.