Before the book gets into the spell descriptions for druids, it goes into the use of mistletoe (or holly and oak leaves) as a druidic holy symbol. Every druid spell with material components requires one of these plants, and the type used affects the range, duration and area of the spell (and possibly saving throws in some cases). Oak leaves are the weakest (reducing said factors to 50% normal), then holly (about 75%), then borrowed mistletoe (that which hasn't been personally harvested by the caster), then lesser mistletoe. To get the full effect of a spell the druid requires greater mistletoe, which must be harvested at Midsummer's Eve with a gold or silver sickle, and caught in a bowl before it touches the ground. (This is all great flavour, but it sounds like hell to adjudicate in the game. I doubt the rule gets used much; it's the first time I've ever seen or heard about it.)
And now, the spells. Druids in AD&D get twelve 1st-level spells, up from six in OD&D. Those six were predict weather, locate animals, detect snares & pits, detect magic, purify water and faerie fire; all of them have carried over to AD&D. In addition they get five all-new spells (animal friendship, entangle, invisibility to animals, pass without trace, shillelagh) and one ported over from the 2nd-level cleric spell list (speak with animals).
Animal Friendship: Allows the caster to befriend any animal of semi-intelligence or less. The animal will follow the caster around, and can be taught three simple tricks per point of Intelligence. This training takes a week per task, and must be done within three months of casting the spell. The druid can so befriend more than one animal, but is limited to a total Hit Dice of no more than twice his level. The spell is basically permanent, which pretty much guarantees any starting druid character at least one or two pets/sidekicks. (Once again, though, the Dinosaur Quandary rears its head: is a dinosaur an animal for game purposes? Game balance says no, but Rad's Law* says yes.)
*Rad's Law: If it's rad, go with it.
Detect Magic: It's the same as the cleric version of the spell, only it lasts a little longer, has a slightly longer area of effect, and takes less time to cast. (It's range and duration are less than they were in OD&D.)
Detect Snares & Pits: When in "the underground" (which I assume to be another word for a dungeon), it detects only simple pits and no other types of trap. Outdoors it detects all forms of trap, with specific examples given being deadfalls, missile traps, and snares. The OD&D version of the spell would only function outdoors, but it had a duration of over an hour, whereas the AD&D version has a duration of 4 rounds per level.
Entangle: Causes any nearby plants to wrap around and immobilise any creatures within a 4" diameter for 1 turn. Targets that make their saving throws can still move at half their normal movement. (No consideration is given here for the size and strength of any affected creature, so things are left to the DM's discretion here.)
Faerie Fire: Outlines multiple creatures or objects with a glowing light (the higher the caster's level, the more targets can be affected). The glow doesn't harm the target, but makes them easier to see and hit when in the dark. (The OD&D spell seemed to only affect one target, and gave no explicit mechanical advantages.)
Invisibility to Animals: Makes one creature touched completely undetectable by normal animals with an Intelligence under 6. Finally, we get a definition of what a normal animal is: "Normal animals include giant-sized varieties, but it excludes any with magical abilities or powers". The tyrannosaurus rex is specifically called out as being affected by the spell, and so the Dinosaur Quandary is solved (and just five paragraphs after I named it).
Locate Animals: The caster faces a direction and concentrates on a type of animal, and the spell tells him if it is with range. The OD&D spell was basically the same.
Pass Without Trace: The target can move through any terrain, and leaves behind no tracks or scent. It should be noted, though, that he does leave behind a trail of magic, and could still be tracked that way.
Predict Weather: The druid knows the exact weather conditions within 9 square miles, for a length of time equal to 2 hours per caster level. The OD&D version of the spell allowed a flat forecast duration of 12 hours, but had a range of 2 square miles per level, and so gave a much greater spread of knowledge. It was also only 95% accurate, whereas the AD&D spell gives 100% accuracy.
Purify Water: Makes one cubic foot per level of water safe for consumption, and can be reversed to contaminate a like volume. It even works on holy/unholy water. The OD&D version of the spell was exactly like the cleric spell purify food & drink, but worked only on water.
Shillelagh: Aside from being fun to say, this spell transforms any oaken cudgel into a weapon that deals 2-8 damage and has a +1 bonus to hit. Not the best spell, but handy when confronted with monsters that are immune to regular weapons.
Speak With Animals: Works exactly like the cleric version of the spell, but is quicker to cast and has a longer range. The druid also gets it as a 1st-level spell, as opposed to 2nd-level.