Barter (Charisma): Putting points into Barter will cause traders to give you more money for your stuff, while you'll have to pay less for theirs. Don't bother with it. You may start dirt pour, but you'll be plenty rich by the time you're level ten.
Big Guns (Endurance): This skill is obvious - it determines how well you can use Big Guns. Big Guns are good weapons for clearing out packs of enemies. However, ammo for them is extremely expensive and fairly rare, so I don't recommend taking this as your primary form of offense.
Energy Weapons (Perception): Affects how you use lasers guns and such. Energy weapons are powerful and ammo is plentiful throughout, but that also tends to be true of Small Guns as well. I suggest you pick whatever better fits your attributes.
Explosives (Perception): Determines the power of mines and explosives. Nice, but you probably won't use these items enough to get the most out of this skill.
Lockpick (Perception): Lets you pick locks, of course. This skill is required to access certain areas of the game, and if you don't have a high enough skill, your only real hope is to maybe bump it up using Mentats. Level this skill if getting past locked doors if important to you, otherwise forget it.
Medicine (Intelligence): Gives you more bang-for-your-buck from medicines. Useful for nearly every character, though you don't have to max it out.
Melee Weapons (Strength): Turns you into a swordmaster. Or a psycho with a sledgehammer. Melee weapons can be very powerful, but you also have to be in your opponent's face. A good choice for high-endurance characters.
Repair (Intelligence): Allows you to fix your weapons. Very useful, not only because of the cash savings, but also because you can spend more time in the Wasteland before going back to Megaton or Rivet City for a resupply.
Science (Intelligence): Allows you to hack computers. Like Lockpick, there are certain things in the game you simply won't be able to access without this skill. Also lets you hack turrents and use them against enemies.
Small Guns (Agility): Not nearly as puny as the "Small" might make you believe, as this includes shotguns, sniper rifles, and machine guns. This is a good choice for a bread-and-butter combat skill. Don't level this if you're leveling Energy Weapons.
Sneak (Agility): Allows you to pass undetected, and perform very powerful sneak attacks. This skill is absolutely murderous at high levels, and nearly useless if kept low, so make sure you're fully committed once you start dumping points into it.
Speech (Charisma): With this skill, you can sweet-talk many of Fallout's characters. There are quite a few interesting quests and dialogue choices that are only viable if you dump points into this skill.
Unarmed (Endurance): Lets you beat down your enemy with your bare hands. This can be fairly effective, but isn't as powerful overall as the gun skills. Good as a secondary skill, in case you are low on ammo.