Assuming an exchange like Coinbase is selling "ETH" for $300USD:1ETH, and I give them $300 then how many of what exactly do I have?
Do I have one wei? http://ethdocs.org/en/latest/ether.html
No, you will have 1 ether (1e18 wei) if you buy 1 ETH on a typical exchange.
A wei is the smallest divisible unit of Ethereum, and so rather than express ether in decimals, it can be expressed in its smallest individual unit, which is a wei. 1e18 wei make up one ether and users choose to add in the decimal point to express amounts in ether (e.g. 4.2069 ETH).
This means 1 ether has 18 "decimal" points.