Curious:

Okay guys, before I ask the question I would just like y'all to know that I come from a place which is reallllly conservative, so I don't know many things, anything I ask is purely because I would like to know more about the topic, it's not cuz I'm berating anyone or anything like that.

So here's the thing the other day we had a society class and our teacher asks us about transgenders and many people answered and most of them were wrong ( yes and that's the reason we have society class ) . So till now I thought that transgenders are ppl who were born with organs(?) Of both the sexes. But my teacher said that it's wrong and that transgenders are males who have become females through surgery . I did Google but idk I got mixed results. In the place where I live transgenders are heavily discriminated, so I don't get why someone would specifically wanna change their body for that.... To elucidate: So when I googled stuff I read that it's a biological thing where they feel that they should have been born a girl, and thus they wanna dress in skirts and stuff, but I don't get it cuz at the end of the day isn't it society which has made all these rules about only girls wearing skirts and stuff like that. ? Does it matter so much that you will have to flip your whole life upside down for it?(yes it's really bad here, they usually don't get jobs even if they're qualified so they resort to begging, it is unfortunate but many ppl are trying to change it)
Okay guys again, keep in mind that I'm only asking because I would like to understand. It's so that if one day I do encounter a transgender, I would not come across as rude.
Also are there female to male trans as well ? Do you have to go thru surgery to be a trans ? Or just dressing as the opposite sex would do ? If the previous question is true then would boyish girls also be trans ?