you should learn it as a phrase, it means it hurts, i have a pain.
can is complicated to translate, we use it in many contexts, it is something like a soul. Although it is a Persian word, we use it even more extensively.
For example in this phrase we rather talk about physical pain. Literally we could say it is "my soul is hurting".
You can use it describe the pain from that, but you would normally use the verb "yakmak" to actually describe a physical burn. "a burning sensation" could be this though, although we may not include it to prevent confusion about the meaning
"I feel sorry" cannot be this. That would be "üzüldüm"
In Turkish there is a term "kırılıyorum" that literally translates as "I am breaking" and is used to describe the aching sensation often felt when one is sick. Otherwise, acıyor, sanciyor or ağrıyor would be used to describe aches and pains in various parts of the body. Eg. Dişim ağrıyor (I have a tooth ache) or Bileyimi kırdım, çok aciyor (I broke my wrist, it hurts a lot). I hope this helps answer your question.