Copper Weight: it defines the number of ounce copper on 1 sq. ft. area, such as 0.5, 1.0, 2.0, 3.0 oz, etc. Usually this parameter is used to specify the thickness of the copper on each layer of the board. It is very easy to work out the thickness of 1.0 oz copper on 1 sq. ft. board based on the copper density (8.9 g/cm3), which is 34.29 µm and rounded up to 35 µm. So we have the following list
In general a copper thickness of 35 µm (1oz) is used on outer and inner layers.
If you need to increase the current capacity of a trace you can either increase the trace width or increase the trace height. If you have 10amps go through your trace, the trace width would become big, in such a case the trace height is altered.
When you need very smal traces, (o.1mm or smaller) it becomes very difficult to manufacturer such a trace on a 1 oz copper layer (under etching). In such cases the copper weight is lowered to 0.5 oz.
The copper thickness also affects the trace impedance.
Depends on current carrying requirement, feature size, required impedance etc, outlayer copper will increase by approx 35micron during through hole plating as well. I would reccomend specifying copper weights on your documentation for each layer, always balancing the laminates around the centre of the board.
You dont have to use heavier copper on outerlayer, what happens is during the through hole plating process you add copper to the outer layers, so if you start of with all layers 18um copper, after plating the outer layers will be approx 35um copper.
Heavy copper will disipate heat better, and carry more current.
In your case without knowing the circuit, size of tracks or devices etc and current capacity, all I can guess at is all layers start at 1oz and the plating adds another 1oz on top for the outer layers.
On a more technical note, if you have a board where you do have to handle high currents, always put the high current tracks on the outer layers so you get better cooling of the high current tracks.