Learn Spanish (Latin America): Me duele mucho la cabeza hoy.

Doler (to hurt) works just like gustar. Literally, this sentence says 'The head causes pain to me.' Because la cabeza (the head) is the thing causing the pain, it is the actual subject of the sentence. Since la cabeza is an 'it', we use the 'he/she/it' form of the verb: duele.

The English translation says 'my head', but the Spanish says la cabeza (the head). Can I say mi cabeza?

You will be understood, but native speakers rarely say mi cabeza here. In Spanish, when it is already obvious whose body part we are talking about (the me tells us it is your pain), you just use the definite article (el, la, los, las). So, 'my head' is simply la cabeza.

If the dictionary form of the verb is doler, why is there a 'u' in duele?

Doler is a stem-changing verb. When you conjugate it, the o in the middle changes to ue in most forms (like duele for 'it hurts' or duelen for 'they hurt'). You might remember this same o to ue change from earlier verbs like poder (to be able to) or dormir (to sleep).