"Ah, cripes," I hear you say. "I fear that this will be another boring Latin grammar node." Well, not only are you right, but you've provided an example of this kind of clause. Lucky you!.

Let's look at the following two sentences:

Timeo ut veniat.
I fear that he is not coming.

Timeo ne veniat.
I fear that he is coming.

These are fairly straightforward: a verb indicating fear (timeo, "I fear"), a conjunction (either ut or ne), and a verb in the subjunctive mood indicating what is being feared (veniat, "he might come"). The only thing that might throw you off is which conjunction goes with which meaning. The negative conjunction, ne, goes with the positive meaning (the second sentence), while the positive conjunction (ut) goes with the sentence that has the word "not" in it (the first one). What gives? To explain it, let's split our sentences into two:

Timeo. Ut veniat!
I fear. May he come!

Timeo. Ne veniat!
I fear. May he not come!

So we divide each sentence in half: the first half consisting of just an indicative (main) verb, and the second an optative clause (which uses a subjunctive verb and ut or ne to indicate wishing for or against something). In that case, the meaning we wanted to express—we want him to come in the first sentence, and we don't in the second—is preserved. Presumably, over time the division of the thought into two sentences gradually faded away (speakers of Latin were notoriously lazy), leaving us with the clauses of fearing we now know and love.

Other than that, these clauses are pretty simple. The sequence of tenses dictates which verb is in what tense, but that's a topic for a different node.