Lands are considered Permanents in Magic: The Gathering as long as they're on the Battlefield. Any card on the Battlefield is considered a Permanent, and once it's there, it stays there indefinitely until someone removes it or you're out of the game. A Land is one of six Permanent types in the game: Lands, Creatures, Enchantments, Artifacts, Planeswalkers, and Battles. A Land in your Hand, the Graveyard, Exile, or any zone other than the Battlefield is not a Permanent.

Lands Are Permanents... Sometimes

The rules say a Permanent is "a card or Token on the Battlefield." They go on to list the six Permanent types (listed above) and point out that Instants and Sorceries never Enter the Battlefield, and therefore can't be Permanents.

It might help here to define "Battlefield". Often, we think of the Battlefield as our playmat or the area immediately in front of us. "Battlefield", though, is a zone in Magic: a place cards can exist. When one of those six types exists on the Battlefield, it's a Permanent. When it exists somewhere else, it's not a Permanent - so even if you place that Murder on your playmat as you cast it to show the table what's going on (you're attempting to kill someone's Creature), it's not on the Battlefield from a game perspective.

Once it's on the Battlefield, it can be interacted with by anything which targets a Permanent, including something old-school, like Desert Twister, or more modern, like Aether Helix.

Another thing worth noting is playing a Land is a "Special Action" and does not use the Stack. A Land changes zones from your Hand (the first zone, most likely) to the Battlefield (the second zone) directly, as long as you have priority, it's a main phase, and you haven't played a Land yet this turn.

Any corner cases?

Lands are Permanents when they're on the Battlefield, which means that anything which counts Permanents (like Ascend) will count Lands you have on the Battlefield. Additionally, Lands count as Permanents you may sacrifice if you get hit by something with Annihilator.

Most Lands are colorless Permanents. Because color is determined in most cases by the color of Mana used in the casting cost, and Lands don't have casting costs, they are colorless and don't add to the amount of Red or Blue Permanents you control for things like Chroma and Devotion.

Keep in mind in Commander a Land's legality in the deck is determined by the color it produces. That means you can't run a Basic Forest in your mono-Black Commander deck, nor can you run a Land which produces anything other than or colorless.

Dryad Arbor seems to need its own column.

This is a Land Creature. Fortunately, the card spells out most of its quirks.

See that Green dot next to the type line, where it says "Land Creature?" That's how they determine color identity for something which doesn't have a color. Dryad Arbor is Green all the time, which means it counts as a Green Permanent for things like Devotion.

It also clarifies it's not a spell, so it doesn't use the Stack - playing it is a Special Action and uses the same processes and limitations of other Lands (one per turn, only Main Phase). It is, however, affected by Summoning Sickness, so it cannot be tapped for Mana the turn it comes into play (unless you grant it Haste somehow!).

Lands which turn into Creatures very often gain color when they are Activated. If you turn Faerie Conclave into its Creature form, it becomes a Blue Creature, which will then count toward Devotion to Blue. Once it stops being a Creature, though, it turns back into a colorless Land - and it never stops being a permanent.

Final Answer

Lands are Permanents in MTG as long as they're on the Battlefield. In all other zones they are not. They are almost always colorless, though they're limited by colors they produce in Commander. They count toward things like Devotion, can be Sacrificed to things like Annihilator, and destroyed by things which say "Destroy target Permanent"