I went to a library means that you went to an, as yet, unspecified library. The other person doesn't know which library you are talking about yet. After this, you would call it the library. Maybe you're visiting a different city and you went to one of its libraries. I went to the library means a specific library which the other person understands that you mean. Maybe you're a student at university and you're talking about the library on campus.

Yesterday not yearstoday.....

I'm going to stick my neck out and disagree with everyone who has given their thoughts so far. "I went to the library" - the emphasis is on what you did in the library, or the reasons why you went there, etc. We assume that, wherever you are, there is a library fairly close by, and no-one really cares which library it is. "I was in London the other day, and I had a spare fifteen minutes so I just popped into the library..." No-one knows which of the hundreds of libraries it was in London, and no-one cares. The topic of conversation is what you were doing in the library. "I went to a library" - the emphasis is on that library itself and the fact I went in. Perhaps it was an interesting building (like the one in the picture), or they had an interesting event going on, or perhaps it was the first time I've ever been into a library. I hope that helps. The two sentences are very different but it is quite difficult to pin down exactly what that difference is.

I can't say there's any strict rule when "the" is better than "a", in these kinds of sentences. We generally just base it on experience - whatever is commonly said, is what we say. We just "follow the trend". But there are exceptions to this following of the trend. Indeed, there are times when one option is necessary and the other is wrong. For example, if they can guess which library you mean, use "the". If you want them to try guessing which particular library you mean, use "a". Anything in between, is based on trend, which you can only know from experience hearing it said by native speakers many times.

It is very simple: A means ( 1 ) The: means ONLY 1! For instance, you have ten pens. I have no pen. Then i will ask you for *A* pen. *Can you borrow me A pen since you have ten*? *The, meaning only one: let us go to THE park, that means there is only 1 park. This is the best way to describe the difference between A and The