How do you use noble gas notations?

The noble gas notations are just used to save you some writing.  For example (for neutral atoms), the electron configurations for the first five atoms are:
H: 1s1
He: 1s2
Li: 1s2 2s1
Be: 1s2 2s2
B: 1s2 s2s 2p1

So since everything after He contains 1s2, if you just substitute [He] for 1s2, you save some writing.
For example,

B: [He] 2s2 2p1

This may not seem signficant with [He], but as you subsitute in higher noble gas configurations you save a lot of writing.  For example,

Ne: 1s2 2s2 2p6

Na: 1s2 2s2 2p6 3s1  or [Ne] 3s1

Kr: 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6
and you can then substitute that in for other elements

So
Rb: [Kr] 5s1
Sr: [Kr] 5s2
etc. etc.

Similarly, once you form ions you still save a lot of writing.

Rb+ : [Kr]
Sr+ : [Kr] 5s1
Sr++: [Kr]

Since things like to go to Noble Gas configurations, you can often use that to try and figure out what oxidation states elements are likely to form.

you would look to find out what noble gas is on the previous row of the periodic table. (a row being a line of symbols that go from left to right rather than going from top to bottom.) then you would start working your way towards the symbol at hand.

Instead of writing the whole set of electrons (1s 2s 2p 3s 3p 3d..etc) you just take the first noble gas that is before your element in the table (i.e If your element is Ba, the closest noble gas would be Xe) and then just write the electrons that come after that(i.e. Ba=[Xe] 6s²). This is very helpful to save time writing if your element has a lot of electrons.