The owner of a call has the right to buy the underlying at the strike price any time before expiration.
The seller of that call has the obligation to sell the underlying at the strike price if an owner exercises the call (the seller is assigned).
The owner of a put has the right to sell the underlying at the strike price any time before expiration.
The seller of that put has the obligation to buy the underlying at the strike price if an owner exercises the put (the seller is assigned).
No matter how you cut it, you have to learn the above details. Perhaps this anecdotal explanation might help:
In common parlance, when someone says, "I put it to him", it generally means that you sent something their way. So if the owner exercises a long put, he's putting the stock to someone else, making him buy it.
In finance, the word call generally refers to getting something. For example, after 5 years most preferred stocks are callable by the company. In other words, they call/redeem (buy) the shares back for the issue price. For options, the owner of the call is saying something similar - I'm calling (buying) your shares at the agreed upon contract prices.
I would suggest that while learning this, you focus on calls only. When you understand them clearly, then turn to puts because for the most part, they're just the mirror image of calls but in the opposite direction.