Credit Card help!?

Many articles say you should pay off you credit cards in full every monthly billing cycle. However, I thought credit is built by paying interest. So say you have a $1000 credit for one credit card. I now make $250 in purchases using that card and I pay off $100 for that month. The company sees I made a payment, they add interest to whats remaining and my credit score improves since I have made a payment for that month.

Now I owe $150 + interest and I leave that $150 for the next monthly cycle. I will have a minimum payment due but now I will pay it off fully and I don't make any purchases using that card for that month. So the following month after that I will have borrowed $0 in credit and therefore that should mean I don't build any credit history, correct? 

I could be completely wrong but I just want to be sure how I can build credit properly. Another hypothetical is say I have made purchases in the same month using one credit card, paid it off then bought more stuff and I do not pay this amount off. This then transitions over to the next month with interest?