Does ごご mean P.M. and afternoon?
Does ごご just mean P.M. as in the time on a clock, or does it also mean "afternoon"? Would you use ごご in the same way as "afternoon", like if you wanted to say "The weather is nice this afternoon"?

Question

Answer

In addition to answers of Ryoko-san and Nao-san, I 'd like to say this.

日本語 Japanese
正午noon＝午前12時＝午後零時
英語English
noon＝12 P.M.

日本語
午後12時＝午前零時
英語
12 A.M.

Therefore we(Japanese) must be careful when we promise with American. We might think that the time is "午前12時" when the American say "12 A.M. ".

Answer

We use ごご for both ways. 
As an additional point, when we use the pm in the sentence, the time is just after ごご. In other word, when you find ごご without any following words related to the time, this means afternoon.

And, yes! You can say that. I will put an example here. ごぜんははれてましたが、ごごはあめですね。

Thanks,

Answer

As you already know, there are 12 hours in AM and PM, it's the very correct way of saying the time. 
However, we use ごご for "afternoon" in everyday conversation. (Ex. 午前中はいそがしいので、午後一時に電話をします[ごぜんちゅうはいそがしいので、ごごいちじにでんわをします]。= Since I'm busy in the morning, I'll call you at 1:00 in the afternoon.) 
We also use あさ(early in the morning)、ごぜん(in the morning)、おひる(at noon / around noon)、ゆうがた(in the evening)、よる(at night) in everyday conversation but at the station, airport, etc, or when time must be said officially, they always use AM(ごぜん) and PM(ごご) for the time of transportation.

I hope this helps you.