What it means
EUR/USD at 1.1000 means one euro costs 1.10 US dollars. Majors pair the dollar with another big currency, crosses leave the dollar out, and exotics include an emerging-market currency with wider spreads.
Example
Buying EUR/USD means going long euros and short dollars at the same time.
In practice
The first currency is the base and the second is the quote. If you think the euro will strengthen against the dollar you buy EUR/USD; if you think it will weaken you sell it. Every forex trade is a view on two economies at once.
Majors such as EUR/USD, GBP/USD and USD/JPY have the tightest spreads and the most reliable liquidity, which is why most traders start there. A pair is most active when the financial centres of both its currencies are open.
What to watch for
Pairs that share a currency move together. Being long EUR/USD and long GBP/USD is close to one large bet against the dollar, so count the combined risk, not each trade on its own.