What is futures price?

I see from financial news that spot price of object A is 100$ and first of March futures price of object A is 150$. What does it mean exactly that futures price is 150$ ? Is it the price of the contract?

Not exactly. It is the contract price that you can enter into, meaning that you can enter into a contract (for free) to buy A in March for $150. You don't have to pay anything upfront (except whatever margin your broker requires, which counts towards the $150)

Does it mean that if I pay 150$ now I will be delivered object A at March?

No - it means if you still hold the contract when it matures in March, that you must pay $150 for A (and take delivery if it's a physical commodity).

I want to enter into agreement with someone to buy object A on March first at 120$, how much do I need to pay to enter into this contract?

Just enough to meet your broker's margin requirements. Since the price of A can go against you, it's possible that you'd have to pay to get out of the contract before March or be forced to buy A for more than its market value. So brokers require "margin" as a collateral to make sure you don't break the contract because you don't have the funds to honor it.

EDIT:

I just realized that the price you want to enter is different than the futures quote. You don't get to pick your price on a futures contract. You have to enter in at the futures quote (or hope that it goes down). You could probably enter into an over-the-counter forward where you pay $30 now and $120 at delivery, but price-wise it is the same (with possibly some differences in accounting).

What you could do is buy a call option with a strike of $120, but that will cost you more than the $30 because you're protected from the price going below $120 but benefit if the price goes above $120. You basically set a "ceiling" for your price. But that's a longer answer...