Improving Your Writing
Reading aloud can be a great help when trying to craft well-made sentences. Most people can improve their writing more when they read it aloud than when they read it to themselves silently.
An incomplete sentence is usually one which is lacking a subject (= what or who is doing the action) and/or a verb (= the action). We use incomplete sentences all the time in speech, and in informal writing, and they can be very effective. However, in academic writing they give an impression of sloppiness and lack of seriousness.
You may well find some academic writing has very long sentences. Indeed students sometimes try to imitate this style, thinking that the more complicated their writing is, the more impressed their marker will be. This is a mistaken assumption. When you are editing your work (but not necessarily when writing a first draft) try to keep sentences from rambling on.
Stray Bits and Structure
When you are editing, look out for what grammarians call dangling articles and dangling modifiers: these are stray bits of sentences that need to go in a different place in order to make sense. Watch out also for the comma splice: this is when a writer uses a comma to link two pieces of language that should instead be separated by a full stop or an explaining word.