Only partially same.

American Style options got two: Calls and Puts

The calls part is almost the same like call warrant but not quite. The call warrants are issued by the company, much longer expiration but American call options are issued by the exchange (eg. CBOE) shorter time frame..(weeks, months and maybe 1 or 2 year aka. Leaps).

Puts are bear side play for Calls.

American Options allows you to make money whether the underlying stock moves UP, Down or Nowhere as long as you know
what direction it will go in the given timeframe.

Of course.. that's what the premium comes in. high volatility = high premium.

high - use selling strategy (covered calls, naked puts, straddles, etc).
low - use buy strategy (naked calls, puts, calls-spread, puts spread).

Time (the expiration) is your enemy and if you're buyer/owner.
But, time is your friend if you're the writer/seller.
Use to your advantage! 

Most people never wants to exercise the options.. they just want to trade it (profits from the premium as the stock price moves up or time decay).

Sometimes exercise may be viable if you think you can wait it out if the trade goes against you or
if you wish to own or get rid of the underlying stock anyway.

Before you play options, you must know the direction and price action of the underlying stock very well.

Remember, one contract equal 100 units of the underlying stock (except in mini-option, 1 contract  = 10 units)
Don't buy big big (eg. 100 calls of AAPL), you could end up with 10000 shares of AAPL later if stuck  (~ approx USD 4.3 mln).