This is a good question--in most animals, the male and the females look different; and you are right, it's a way for them to tell each other apart. Sometimes it is a subtle difference, like size, and other times it's a big difference, like the examples that you give. Most of the animals probably have these differences as a way to help select a mate. It may be that the deer use their horns for fighting--but what do they fight about? Think about peacocks and peahens (the girl peacocks). They look very different from each other. The peahen probably selects her husband based on how bright and big his tail is. Can you think of other examples?

You made a good connection between what you saw in deer and what you saw in quail.They are both examples of what we call "sexual dimorphism", which is just a way of saying that males and females look very different in some species. This is because female birds and mammals are picky about their mates. A female bird or mammal usually has to care for eggs and/or babies, so females can only have a limited number of offspring. In species where male care of offspring isn't necessary, male animals can mate with several females (if they get a chance).

In California quail, males and females form pairs, so both males and females only have one mate. Normally this would mean that even the poor-quality males would get a mate. Unfortunately for the males, there are always more adult males than females. This seems to be because raising young is dangerous and tiring for females. Whatever the cause, it means that some males are always left out. Males compete with each other for the chance to pair with a female. Because there are lots of males to chose from, the females can be picky. You are right that males are probably advertising themselves by looking different from females. They also have special calls and behaviors to say, "I'm an unmated male." Males that are flashy looking may be able to advertise to females and other males that they are healthy. There's a good site about this topic at: click here

In deer, males don't take care of the young at all. No pairs are formed, so no females have to settle for a weak male. This means that a few males can be the fathers of a lot of offspring. It also means that many males don't get to mate at all. There's a lot of competition between males to be the successful fathers. You're right that males use their antlers to fight with each other. If the antlers were to protect deer against predators, females would have them too, and males would keep their antlers all year instead of shedding them.

If you were a female deer, would it be better for you to have a daughter, who is almost guaranteed a mate but will only have one fawn per year, or have a son, who might have lots of offspring, but is much more likely to have none at all? Would it make a difference if you were healthy or if you were in bad shape? What about if you were a male deer, would sons or daughters be a better choice?

Incidentally, the main difference between horns and antlers is that horns are kept all through an animal's life. Cows, bighorn sheep, and the African antelopes all have horns. Females have horns too, though they tend to be smaller. Antlers are shed each year. Male moose, deer, and elk all have antlers. The only female animals with antlers are caribou.

I guess you are right! You mentioned the two main reasons. Each animal has different shapes because of the evolution of the species over times. Those differences are obvious in between a human being and a fish, an alligator or a mouse, but also in between males and females. Their differences help each of them in being more adapted to do some things or others. For instance, male deers can fight with his horns (for territorial and breeding defense), male quails look more attractive to female quails because of their small crest...
So you were right: (1) females and males can recognize each other because of these differences, and (2) those differences help them in carry out their roles in nature.