However
Some old-school grammar books will tell you not to use 'however' at the beginning of sentence as a connector or transition with the previous sentence. However, most modern grammarians disagree with them. Using 'however,' followed by a comma, clearly means 'nevertheless' and makes a useful connector when you are stating something that disagrees with the previous sentence. In fact, I just used it again in the second sentence of this paragraph. The old-school grammar books would tell me to rewrite that sentence like this:
Most modern grammarians, however, disagree with them.
How is that better? The 'however' now breaks up the main idea of the sentence, and it isn't clear right away that I'm disagreeing with the previous sentence. However, putting 'however' at the beginning of a sentence makes it serve as a 1-word introductory phrase, a group of words that serves as a transition or sets the stage for the main clause of a sentence, such as:
In Arizona, the population has grown exponentially over the past several decades.
That's why the comma is important, as you always want to put a comma between your introductory phrase and the main clause of your sentence.
Because
A similar dynamic is at work when using 'because.' Grammar teachers often say not to use 'because' at the beginning of sentence because they think you'll do this:
Because the population of Arizona has grown exponentially.
This is a sentence fragment because the 'because' has made this a subordinate clause, a group of words that includes a subject and verb but can't stand alone as a sentence. The 'because' implies some sort of cause and effect, but there is none described here, so it's just left hanging. However, subordinate clauses can be used as introductory clauses, which, as the name implies, are pretty similar to introductory phrases. So check this out:
Because the population of Arizona has grown exponentially, the state is facing a severe water shortage.
The part after the comma is an independent clause, which can stand on its own as a complete sentence, so now our 'because' clause is serving to introduce it.