Asgard is the home of the Gods, or at least one tribe of the Gods, the Aesir. It is also home to some Vanir Gods, namely Njörd, Freya and Frey, as well as to Gerd and Skadi, who are of Jötun descent. These days Asgard is often understood to be synonymous with Valhalla but that is just not the case. Asgard is also not a kind of heaven where there is no pain and suffering but simply the world where the Gods have their homes.
Each God has a realm within Asgard and a hall within that realm. Odin’s realm is called Gladsheim (bright home) for example and even has two different halls, Valaskjalf (Bench of the Slain) and Valhalla (Hall of the Slain). The other Gods and Goddesses have their own realms in Asgard and since Odin as well as Freya are able to invite fallen warriors into their respective halls the same might be true for the other Gods as well. Frigg who is closely connected to motherhood may welcome mothers that died in childbed while Ullr could be welcoming the best hunters into his wild realm. This is only speculation though as there are no surviving sources that would confirm this.
Asgard isn’t a safe place. It is connected to Midgard, the world of men, through Bifröst, the rainbowbridge that can be seen in the sky from time to time. The red color in the rainbow is believed to be made out of fire which prevents the Frost- and Hill-Giants from entering Asgard uninvited. Since this doesn’t always work the Aesir also have a guardian named Heimdall who has his hall right next to Bifröst and watches it day and night. But even that was not enough for the Gods to feel truly safe and so they built a huge wall around Asgard as well. The fact that Asgard isn’t safe is actually one of the most fascinating aspects of Norse Mythology as it shows how different the idea of divinity was at the time. The Gods aren’t omniscient, omnipotent or even immortal. It also speaks to the power and vigor of the Giants that these measures were even necessary in order to keep them from overthrowing the Gods and replace them at the head of the Cosmos. In the end at Ragnarök (link) they actually do win the final battle against the Gods where many of them like Thor and Odin meet their end but they fail at truly staging a coup as it doesn’t seem to lie in their nature to rule and so the whole cycle starts all over again.
It also has to be made clear, that the pantheon we have based on the lore recorded in the Edda’s, is surely incomplete. For one we know that many Gods that were worshipped during the Viking Age had been around for a long time, some of them being mentioned by Tacitus and/or in the Meresburg Charms. But the fact that Wodan or Wuotan are etymologically going back to the same root as the name of Odin, doesn’t automatically mean that the God worshipped under these names was one and the same. A lot of time passed between the Germanic Tribes forcing their way into recorded history with a bang when defeating the Romans in 113 B.C. at the Battle of Noirea and the late Middle Ages when most of the Myths were finally put on paper. Aside from that many deities simply disappeared over time. The Goddess Nerthus for example that according to Tacitus was a very important deity at the time, is simply absent in the Norse sources. There is a theory that she simply changed sex and lived on as Njörd based mostly on linguistic evidence but the idea has lost a lot of traction over the last few years. The national God of the mainland Saxons, Saxnot, is also one of the Gods that got lost in the mists of time and there will have been many more throughout the ages.