Buy to Open means that you are purchasing a call or a put . When you buy to open, you are opening a new long position in the option. You own the option, and you can control whether you exercise it or not. You pay the premium for the option.
Sell to Open means that you are writing a call or a put. When you sell to open, you are opening a new short position in the option. As the writer of an option, you are entitled to the premium that the buyer paid for the option. You are obligated to buy or sell shares of the underlying security if it is exercised.