Learn Italian: Penso a mia madre. - I think about my mother.

In Italian, the verb pensare (to think) is followed by the preposition a (to) when you are directing your thoughts toward someone or something. Literally, you are saying 'I think to my mother'.

Why is there no la before mia madre?

In Italian, you usually put the definite article before a possessive word, like in la mia macchina (my car). However, you must drop the article when talking about a singular, close family member. Therefore, it is just mia madre, not la mia madre.

Can I add the word io and say Io penso a mia madre?

Yes, you can add io (I), but it is optional. The -o ending on penso already tells us that 'I' am doing the thinking. Italians usually leave the subject pronoun out unless they want to emphasize exactly who is doing the action.

How does pensare (to think) differ from the verb sapere (to know) that we are learning right now?

You use pensare when you are having a thought or opinion. You use sapere when you know a concrete fact or a skill. Keep in mind that if you wanted to say 'I know my mother' (as a person), you wouldn't use sapere at all! You would use conoscere, which is the verb for being familiar with people or places.