Verb tenses in English are somehow easier than most other languages - the peoples who crafted the English language somehow realized that the subjunctive tense (a way of changing the verb to imply a possibility rather than a definitive) would be easier if they used other words to signify it, instead of a change in the verb itself. (German, the English parent language, retains subjunctive verb tenses, as do most of the Romantic languages.)
Our future and perfect tenses are also signified by an addition of a word, instead of a word ending or other change. Adding the word "will" to a verb phrase creates the future tense (I will go to the store tomorrow), while a form of "to have" creates a perfect tense (Future: I have will completed my homework by tonight, Present: The dog has barked at the mailman every day this week, or Past: My daughter had gotten the flu before but that time was worse.) But the perfect tenses are a whole other can of worms, so let's keep going.
The most common verb tense error is when the tenses don't match. In general, this occurs between past and present verb tenses (the ones most reliant on endings), but can occur between other tenses, as well. The problem with this grammar mistake is that it is a logical one, not a simple rule, which is why it happens so often.
- Danny is going to the store an hour ago. Obviously, Danny can't currently be going to store an hour ago - he went to the store (yay, irregular verbs!).
- Sandra jogged at the beach while listening to the Beatles every day. Here, we have a tense shift: If Sandra is jogging every day, she does so in the present sense. That "every day" forces a present tense verb - meaning that "jogged" should be "jogs".
- Before we arrived at the movies, John will call to say he would be late. Notice here the temporal (timing) clues that the sentence is giving you, dictating what the tense should be. The "we" must already be at the movies, because they already know that John has called. Therefore, we need to say "Before we arrived", in the past tense. John, too, has already placed that call - this isn't existential questioning or time-travel - so "will call" should be "called".
Notice, however, in that last example that when you start playing with tenses and timing, it is possible to come up with some creative uses. In fact, many fiction writers will do this as a type of style (the Modernists, for one, loved to play with verb tense). In grammar, though, the rule is always the same: your sentence must make sense to the reader. Most of the time, a verb tense confusion is not on purpose for style - it's just a simple mistake.