Gogogo (ゴゴゴ) is the sound of a deep rumble. In manga it means a heavy, menacing aura: something big is about to happen.
Two rivals face each other. Nobody moves. Behind them, huge jagged letters fill the panel: ゴゴゴゴゴ. You cannot hear anything, but you can feel it.
What it means
ゴ is a low, heavy sound, and repeating it makes a rumble that will not stop: the ground shaking, a storm on the way. In manga it became the sound of tension itself. Someone powerful has arrived, or someone is very, very angry.
Where the meme comes from
The manga JoJo's Bizarre Adventure made it famous: its tense scenes are full of giant ゴゴゴ and ドドド in the background. English-speaking fans call it the “menacing” aura, and today the letters turn up in memes, jokes and other manga everywhere.
How to use it
ゴゴゴ is mostly written, not said. In a chat with friends, people add it after a message to show they are angry or deadly serious, half as a joke. Said out loud, slowly and low, it gets a laugh.
ゆきさん、ぼくの ケーキを たべましたか。
Yuki, did you eat my cake?
…はい。たべました。
...Yes. I did.
ゴゴゴゴ…
*menacing rumble*
Close cousins
ドドド is the same idea, even heavier: something charging forward, or a presence you cannot ignore. ゴロゴロ is a rolling rumble with nothing scary in it: thunder far away, or a purring cat. ザワザワ is a crowd murmuring nervously, the sound of everyone noticing at once.
In manga and anime
In manga the letters are drawn big and jagged, often behind a character who has not even moved. In anime the same moment usually gets a low rumbling sound and a slow zoom on someone's face.
Can you use it?
As a joke, yes: ゴゴゴ after a message to a friend reads as playful anger. It is not a word for serious talk, and in a polite email it would look very strange.