Option Price
The price of the option depends on the following:

Asset Price:
For an American or European call option the higher the asset price the higher will be the option price all other things unchanged. On the other hand, for a put option the lower the asset price the higher will be the option price all other things being held constant.

Exercise Price:
For an American and European call option the higher the exercise price the lower will be the value of the option all other things being held constant. The opposite is true for the put option when the higher the asset price is the higher will be the value of the exercise price is the higher will be the value of the option all other things unchanged.

Interest Rates
An increase in interest rates lowers the present value of the exercise price. For an American or European call option this means the cash outflow is lower in discounted terms, which in turn makes the option more valuable to the buyer of the option, other things being equal. The opposite is true for a put option as the cash inflow is lower in discounted terms when interest rates increase, making the option less valuable to the buyer of the option, other things remaining unchanged.

Volatility of the Asset Price
Volatility is a friend for option contracts. When expected volatility is high there is a greater likelihood that the option price will rise (or fall) in relation to the exercise price hence increasing the value of the American or European call (or put) option more valuable, other thing kept constant.

Time to expiration
When the time to maturity is farther away in the future the American or European call and put option prices will be higher in general. This is because:
 There is greater likelihood that the prices will move sufficiently to breach the exercise price and provide value to the option buyer.
 For call options the cash outflows (from the exercise price) are discounted over a longer time making them lower and thus more valuable to the buyer. For American put options even though this means lower cash in flows in discounted terms the benefit from (1) usually tends to outweigh the latter and if this is not the case, they have the option of exercising the option earlier than the maturity date.
For European put options a longer time to maturity can go either way depending on which of (1) or (2) has a greater impact.

Cash Distributions
This is relevant if the asset is a common stock that pays cash dividends. When a cash dividend it paid the stock prices falls. For a call option this means that the value of the option will decline all other things being equal. For a put option the opposite is true.
European option price
The option prices calculated using BlackScholes formulas are as follows:
Call option price (c)
Put option price (p)
Where
N(x) is the cumulative probability distribution function (pdf) for a standardized normal distribution
S_{0} is the price of the underlying asset at time zero
K is the strike or exercise price
r is the continuously compounded risk free rate
? is the volatility of the asset price
T is the time to maturity of the option
q is the yield rate on the underlying asset. Alternatively, if the asset provides cash income, instead of a yield, q will be set to zero in the formula and the present value of the cash income during the life of the option will be subtracted from S_{0}.