Those sentence types are common terms from the AP English Language and English Literature exams. So, if you can master not only analyzing them but also writing them now, you will be in good shape as a writer.
1. Loose or cumulative sentence: These two sentences rely on the same technique; they begin with the main independent clause and then pile on information in phrases and dependent clauses. If you were to place a period after the initial independent clause, it would be correctly punctuated as a sentence. Below is an example from writer Terry Tempest Williams.
"The women moved through the street as winged messengers, twirling around each other in slow motion, peeking inside homes and watching the easy sleep of men and women."
Notice that the bold part could stand by itself as a sentence, but the rest of the phrases could not.
2. Periodic sentence: A periodic sentence is just the opposite of a loose/cumulative sentences. It begins with phrases or dependent clauses and holds off on incorporating the main independent clause until the end. It cannot grammatically end correctly before the period. Below is an example from Ralph Waldo Emerson.
"Crossing a bare common, in snow puddles, at twilight, under a clouded sky, without having in my thoughts any occurrence of good fortune, I have enjoyed a perfect exhilaration."
You can see the difference between the periodic sentence above (notice the bold independent clause at the end) and the loose sentence structure. If you were to leave off the bold portion of the above sentence, it would be a fragment.
3. Balanced sentence: A balanced sentence is essentially parallelism, but sometimes it is more specifically a sentence that has at least two clauses which are similar in length and word order. Here's an example about characters from Of Mice and Men.
George is mentally keen and small in body; Lennie is mentally simple and large in body.