The most common problem in sentence creation is when you're not actually writing a single sentence. Instead, you either end up with a half-sentence, called a sentence fragment, or multiple sentences in one, called a run-on sentence. When editing, the goal is to catch as many of these as possible, as they tend to can stick out like sore thumbs in otherwise great writing.
Sentences need two things to be complete: a verb and a noun to take the verb's action, called a subject. A sentence fragment has only one or the other.
The main reason this occurs is that there are still other things in the sentence - adjectives, adverbs, a verb that doesn't work with the noun that's present so that noun can't be the subject. When editing for sentence structures, you need to ask yourself, "Where is this sentence's subject and verb?". If they're there, you have a complete sentence.
We often speak in sentence fragments. When writing dialogue or when using a very close point of view to a character, you can often get away with using sentence fragments infrequently. However, when writing academic or more formal writing, sentence fragments never belong. The writing tip "Write like you speak" only goes so far.
Run-on sentences occur when a writer is trying to cram too much thought or creativity inside a single sentence. The very best writers (we're talking Pulitizer-prize winners here) can create sentences that last for pages that really aren't run-on sentences. The normal person, though? They should be wary of any sentence that lasts over four typed lines.
When looking for run-on sentences, you're actually looking for a cousin of the sentence fragment. In a run-on, you have too much information without it being properly structured. In grammar, everything has a specific place to go and if you have multiple subjects or multiple active verbs, there needs to be some connecting words, like conjunctions (and, but, or). Without that connection, you have a run-on.
You can fix the run-on by breaking the sentence apart. In general, for most writing, it's better to be clean and precise. Even if you're not sure you have a run-on but you definitely have a very long sentence that seems to ramble on, you should probably just break the sentences down into smaller sentences, anyways. Just make sure you rearrange things so you don't end up with a sentence fragment by doing so. Smaller, more compact sentences tend to be easier to read.