Is each realm its own planet?

All the realms are bounded to one giant tree called Ygdrassil.

The story has changed with time, all of Thor's world is based on old european Norse mythology. Thor's tales are from a time when Europeans believed in a land of faeries, creatures that come from clouds, winged horses and dwarves. There was no 'math' or 'science' but some of the old legends maybe had stories of mysticism and alchemy.....that's about as technical as it ever got, people at this time did not understand geology full or know of planets or math, everything was about lands of magic and a vast sea of space, his story is very ancient, very tribal.

Thor and Earth and everywhere was thought to belong to the mystical cosmic ash tree, Yggdrasill, its a big tree in space, the trees legs go way down below the Earth down under the earth and under space itself and into the land of the dead and the land of fire demons, its arms go up into space. The Tree grows and grows into the stars it goes up above Earth into space clouds with all these stars as leaves and its branches can connect one planet to another, the north of the tree was cold, its lands icey, its mountains had snow, its rivers frozen, to the south of the tree the lands were hotter, dryer, they had volcanoes. The people of the time explained everything through magic reasoning, the death of Ymir helped make the world his blood made lakes on earth, his brains became clouds, his hair transformed into trees etc....that's about as much as I understood.

When Stan and Jack took the concept and made him a Marvel characters they transformed the Nine Realms into something more tangible, it became gateways and entrances to another dimension. The Asgardians were still mystical and magical people but they were scifi-ish, their stories sometimes became grounded in science or their knowledge was explained by Asgardians being an advanced alien civilization.

J. Michael Straczynski and Kenneth Branagh ground the Thor character even more, now he's in space itself with spectacularly detailed fx. He lived in a world which can be seen in Hubble telescope pictures, the realms are no longer mystical kingdoms or far away magic lands instead they are planets now connected by a stargate /startrek style rainbow bridge. The gas clouds/Nebulas and stars in the galaxy form a giant galactic TREE and the Milky Way is but a branch of that tree. Thor does still retain that magical mystical quality about him but all of Asgards feats are explained through it being an advanced technological place.

I always figured the Nine Realms were nine different dimensions where certain planets exist like Nidavellir, Muspelheim and Jotunheim. For instance, Midgard (to me) is the realm of our universe, not just Earth itself.

Are you talking about the comics or the movies?

In the movies each realm is its own planet with Asgard being some sort of very large asteroid. The Aesir are not real gods, but advanced aliens who use some form of science so far beyond human understanding it might as well be magic. They even call it magic a few times. A few lines have implied Asgard itself if not each realm may dwell in a different dimension with the entire universe of Earth being a dimension separate from the others. Yggdrasill appears to be some sort of galactic cluster or grouping.

In the comics, the Asgardians are gods and have always been gods. In Marvel, there is a class of beings treated as gods who use magic not science. Asgardians are a group of these. In the comics, each realm is a separate dimension though some writers have treated them all as one with Earth being the only separate one. Naturally occurring dimensional portals connect the different realms so seamlessly you can ride from one realm to another without noticing you go through a portal. Yggdrasill is what links the realms. Sometimes it is shown as a real tree you can climb up and down to get to the different worlds. Other times it is its own space/time continum that connects the dimensions. It is part of the larger universe, but separate enough that those nine dimensions are their own special grouping separate from other dimensions. Asgard has been shown as its own planet/asteroid at times, but the same cannot be said for the other realms.

In short, Movies: Asgardians are advanced aliens who use science and each realm is its own planet possible residing in its own dimension.

The comics: Asgardians are gods who use magic and each realm is its own dimension with each realm not shown to be its own planet.

well then what kind of "science" allows Loki to create illusions,and shape-shift? or makes Thor's hammer control the weather?

If you are addressing the comics then it is magic.

If it is the movies the whole point is that it is science so far beyond human understanding it might as well be magic. Clark's Third Law and all that. The television Babylon 5 has technomages who readily admitted they used advanced science to simulate the effects of magic so it looked like magic. True Gods are almost always considered magical creatures, but the New Gods use computers and such so it is hard to tell if some of their objects are incredible advanced technology or magical items in the form of technology. The Stargate television series had various aliens claim or misinterpreted as being gods through advanced tech. Granted, they mostly pulled it off by using it against dark age humans, but the principle is still the same. This is why the Bifrost is a wormhole cannon and on an actual bridge. However, I do admit the first film left some wiggle room.

It's interesting though, because Marvel put out a poster hyping Thor the movie with the depiction "The God of Thunder" for Thor, "The Goddess of War" for Sif and so on.

Here's a dressed-up map of the NIne Worlds of Asgard found in the old Official Handbook of the Marvel Universe.