If you are just planning on being a casual cuber then learning a special notation for moves might seem a bit nerdy, or over the top. But it isn't.
Firstly, the notation is really simple. So it isn't much effort.
And secondly, everyone uses it. So any source of help you seek will always use this notation.
Finally, having letters to remember rather than moves is easier for some people...
The official notation is defined by the World Cube Association. It is based on the notation originally proposed by David Singmaster, with some modifications.
L = left
R = right
U = up (the top)
D = down (the bottom)
F = front
B = back
The most important thing to remember is that each move is clockwise if you were looking straight at that face. So an L turn is towards you, but an R turn is away from you. This sounds odd but makes sense if you actually look at it on a cube... trust me :)
Adding a ' means turn the opposite way. And adding a 2 means do two turns.
RR = R2
R'R' = R2
RRR = R'
R'R'R' = R
It's worth pointing out that L2, R2 etc can be done either way. Sometimes you might see R2' which is a hint that R2 anticlockwise might be faster than clockwise, but it is the same move and up to you to decide which you prefer.
A couple of the algorithms I use have wide moves in, but not many. Lw just means turn the left face and the middle section together; and Rw is move the right face and the middle section together. Those are the only 2 wides I use... I think.
There's one final move which isn't official but does appear in some common algorithms. That's the M move... sometimes called an M-slice. Here you move the middle section only, in the same direction as the R face.
(in the official notation an M counts as 2 moves - Rw R'. Unless you are interested in fewest moves competitions then this doesn't matter at all and M is easier to remember).
As a beginner you can learn algorithms and then chant them as you move R U R' U R U2 R'
Later you will find that some moves have names like Sune, Anti-Sune, Sexy-Move; but mostly you will just get used to executing them and will do it from muscle memory without even thinking about the notation any more.
Another place you will see the notation is in all the common timing apps. Usually they will show you a scramble of about 20 moves to solve. It's a good habit to get into using these... just in case you aren't very good at scrambling the cube into good random positions yourself.
Edit: sometimes you will see algorithms containing brackets like this (R U R' U) (R U2 R'). The brackets are just there as a suggestion for grouping, they don't change the actual moves that you do at all. It might be that each group can be done as a single fluid move, or that the groups help you to memorise the algorithm... or both :)
