Cross pairs do not involve the US Dollar. For example: GBP/JPY is a cross, GBP/USD and USD/JPY are not. Beginners often misuse the term "cross pair" or "crosses" to refer to any and all currency pairs in the forex market.
Obviously, this won't affect whether a trader will become profitable or not, but sometimes it helps to be able to communicate with experienced professionals using accurate terms -- especially when a term is actually used to distinguish one thing from another.
Historically, the term was used because the United States dominated the world economy at the time that the current floating rate system was established in foreign exchange. Buying a cross pair like GBP/CHF actually consisted of two simultaneous transactions: buy GBP/USD (long GBP, short USD) and buy USD/CHF (long USD, short CHF). Inherently, the long and short positions on the USD cancelled itself out, so the net position would be a long GBP/CHF position.
For all intents and purposes, the interbank market still works the same way for many cross pairs but there are a few exceptions. (EUR/JPY, for instance, is actually quoted between tier 1 banks without any USD involvement, but it's still considered a cross pair in forex terminology.)
The Yen crosses, EUR/JPY and GBP/JPY, are known for their volatility with large average daily ranges. In contrast, the majors that they are made up of (GBP/USD, EUR/USD, and USD/JPY) are far less volatile on a daily basis -- at least by the forex market's standards.