My 'favourite' weapon is kind of a tricky question, since the weapon that I personally most enjoy using isn't actually the same weapon that I consider to be the 'best' of the loadout weapons.
Prior to the weapon tuning update, my favourite weapon was the DMR, which I also regarded as the 'best' loadout weapon at the time. It had the longest effective range, the fastest kill time of any of the precision weapons (or so I think - the Carbine might have been a little quicker), a large magazine, a powerful scope and fairly strong aim assist. It also had a great firing sound and generally just pretty powerful. The only thing I didn't like was that it was too dominant: you could easily gun down automatic users even at close range and it outperformed virtually all the other precision whether scoped or unscoped, so even though I was disappointed at my favourite gun becoming weaker I'm still glad they nerfed it because it makes the game feel a lot more balanced.
Post-update, my favourite primary weapon is probably the Assault Rifle. Ever since Halo 2 came out I've been wishing for an automatic which felt as good as the Assault Rifle in Halo: CE, nd every time I've been disappointed. Halo 3's version was pretty good: it was reasonably powerful, fairly accurate and combined really well with the melee system, but it didn't feel like anything special and was almost always outclassed by the Battle Rifle, the Carbine or just dual-wielded SMGs or Plasma Rifles. The older model found in Reach was absolutely terrible, being weak, inaccurate, and horribly ineffective thanks to the removal of shield damage bleedthrough, and though the pre-update model in Halo 4 was a step in the right direction, it was still outclassed at anything above short range by the precision weapons and still only existed on even footing at close range, rather than having an actual advantage.
But post-update, I love it. The sound design, muzzle flash, and thick metallic model all combined to make it look powerful anyway, but with the improved kill times and more controllable recoil it's become something truly satisfying: I'm fairly sure if you hit with every shot it's possible to kill two Spartans with a single magazine, and provided you manage your shots you can reliably hit stuff even out to medium range. The extra weapon skins from the Bullseye pack also look amazing: I have a particular fondness for the shark skin, though as I like to match my loadout skins I normally just use the gold-plated one. However, while it's now not just a good weapon but an excellent weapon, I don't really consider it to be the 'best' weapon in the loadouts as even though it has better range than before it's still just not versatile enough. Instead, it's still essentially a specialist weapon like the Lightrifle or the Plasma Pistol: it's much better than the average precision weapon when used at its correct range, but at anything other than that it's just not going to cut it. As a result, while I have it as a backup weapon on one of my loadouts, I still wouldn't consider using it as my sole primary during competitive gameplay.
So as for the overall 'best' weapon, I'm going to say the Carbine. It has a good, powerful sound and a satisfying projectile which makes it fun to use, and some of the alternate skins are also pretty nice, so it just feels good to begin with. But what really elevates it above the Battle Rifle comes down to the fact that it kills very slightly faster (a tenth of a second or something), that it's projectiles don't spread so much, and that because you can pull the trigger more quickly it works better at long ranges against opponents using scopes on things like the DMR and Lightrifle. I think the Battle Rifle is more popular both because of nostalgia and because people tend to feel more comfortable using human weapons (and it might have a little bit more aim assist - not sure on this, but it does feel a little easier to aim), but for my money the Carbine is probably the best primary weapon in the loadouts right now.
As for favourite power weapon - Shotgun all the way. It's horribly short ranged, fires slowly, doesn't hold much ammo, can't be deployed as effectively as older models thanks to larger, more open maps and faster movement, is crippling overspecialised, and is the single most satisfying weapon to use thanks to the cyclying animation after each shot and the firing sound. My God, that firing sound. 'Best' power weapon is the Incineration Cannon because it's stupidly overpowered.