The first example that you give (It started to rain) is the past simple tense. The verb to start is conjugated in the past simple (it follows the regular verb “-ed” ending, which is also known as weak verbs, meaning they do not undergo a change to the stem of the verb). Following this, the verb to rain is left in the infinitive. This is because, when multiple verbs are combined in the past simple tense, the first verb shows the tense (i.e. past simple) and the rest are left in their infinitive forms.
For the second example (It started raining), the verb to start is also in the simple past tense. Afterwards, the word raining, although it looks like a verb, actually acts as a noun. This is known as a gerund. Here are some other examples of gerunds:
- Smoking is bad for you. – The work smoking is actually a noun that comes from the verb form. Smoking is the act (in this case) of lighting a cigar, cigarette or similar thing and putting it into your mouth.
- Hitting other people is not a nice thing to do. – In this case, hitting is the gerund and means the act of punching others.
If you need or want to review tenses, here is A Review of Active Tenses in English