derecha refers to right as a direction, while derecho refers to legal rights or the notion of correctness (the "right way" to do something) derecho can also mean "straight ahead" as a direction, or to stand "upright"
Indeed, at the end of Spanish TV programs you'll see words like "reservados todos los derechos". There is also a straight-line wind called a derecho.
oculos dexter -- right eye. I think derecha probably comes from dexter (Latin, right).
Just a quick q, since this word can be m or f, how do you know when to use either? For example, the people of that country have rights. Would either gender be acceptable?
No, the male and female versions have different meanings (see rspreng's answer above)
Except as an adjective when, of course, it follows the noun eg el pie derecho, la mano derecha.
You would be well served to look at a good dictionary (eg http://www.wordreference.com/es/en/translation.asp).
When talking about multiple people you usually use the M form of the words. Like when saying "them" you say "ellos" even if there are females in the group. Although if its all females in the group, then you use the F form. Choose the right! #CTR