お〜する comes from a form of polite speech in Japanese called
humble speech. This type of speech is used to refer to the speaker and their actions. This is opposed to
honorific language, which refers exclusively to the listener and their actions.
お〜する is almost exactly the same as
お～になる, as in it simply means
する.
お〜する is regularly used with words of Japanese origin, while
ご～する will be used with words of Chinese origin. The
する in this compound will attach directly to the ます stem of almost any verb.
While the
する in this structure will attach directly to the ます stem of verbs, this does not mean that verbs that are already
する verbs will become しする. In the case of
する verbs, simply change the
prefix to ご.
ご登録（とうろく）する方（かた）はこちらをクリックしてください。
For those who want to register, please click here.
ご注文（ちゅうもん）する時（とき）はこちらのベルを押（お）してください。
When you are going to order, please press this bell.
In rare cases, お may be used instead of ご, when attached to words of Chinese origin. There is no specific rule for this, and they must be memorized on a case by case basis.
先輩（せんぱい）にお電話（でんわ）するのがすきです。
I like to call my senpai.
友達（ともだち）とお勉強（べんきょう）するのは楽（たの）しいです。
It is fun to study with my friends.
Thankfully, the words of Chinese origin that use お instead of ご are extremely common verbs. This means that you will be exposed to almost all of them very quickly. All others may be assumed to use ご.
In the same way that
お～になる is a more common variation of
なさる (both
honorific language),
お〜する is a more common variation of
いたす (both
humble speech).
5分後（ふんご）にお電話（でんわ）します。
After 5 minutes, I will call you. (I will call you in five minutes)
5分後（ふんご）にお電話（でんわ）いたします。
After 5 minutes, I will call you. (I will call you in five minutes)