Strength is primarily relevant to two game mechanics: Carry Weight and satisfying the minimum Strength requirements on weapons. You gain 25 lbs. of carry weight per point of Strength (unless you have the Small Frametrait, in which case it's 15). Also, if you meet the minimum strength requirements of your weapon you do not suffer accuracy penalties with that weapon.
The Brotherhood of Steel can perform an operation to increase Strength by 1, assuming the player can afford it (Fallout 1) or has found the appropriate module chip (Fallout 2).
Buffout temporarily increases Strength by 2.
Several Perks increase Strength. Adrenaline Rush raises Strength by 1 when the player has less than 50% health. Gain (Statistic) Perk can be used to boost Strength by 1. Some Perks raise Strength when checking for modifiers. Heave Ho! raises Strength by 2 for purposes of throwing weapons. Weapon Handling boosts raises Strength by 3 for purposes of checking the minimum Strength needed to use a weapon.
The Bruiser Trait raises Strength by 2 but lowers the player's AP.
Strength slightly increases melee damage and carrying capacity. Melee damage is at 1 damage for every 2 strength (+5 damage at 10) and carrying capacity is an added 10 for every point (+100 at 10). Strength does not affect unarmed damage, even though Iron Fist requires some strength. Unlike previous Fallout games, none of the weapons have a minimum strength requirement, and a character with 1 strength can use a minigun without penalties.
High Strength can be used to intimidate others when in a conversation.
Note that while at first glance the damage bonus seems minor at best, the damage bonus is applied after a weapon's base damage has been adjusted by the Melee Weapons skill and weapon condition. So, the damage bonus can be a very significant increase for much of the game. Moreover, even with maximum Melee Weapons skill and weapon condition, high Strength can still amount to a 10% or better increase in the weapon's damage (see specific weapon pages for specifics). This is basically equivalent to or better than a free perk. Moreover, multipliers on damage (from perks like Entomologist) also affect the bonus damage, in case there was any confusion.
Strength in New Vegas modifies Melee damage, Carry weight, Melee Weapons skill and which weapons you're strong enough to use. If you don't meet the strength requirement of a weapon then you will be less accurate (for ranged weapons), or hit slower (for melee/unarmed weapons). You can reduce the strength requirements of weapons by two if you take the Weapon Handling perk.
Similar to Fallout 3, while at first glance the damage bonus seems minor at best, the damage bonus is applied after a weapon's base damage has been adjusted by the Melee Weapons skill and weapon condition. So, the damage bonus can be a very significant increase for much of the game.
In addition, with the changed use of Damage Threshold instead of Damage resistance as an armor mechanic, the bonus damage from Strength can significantly increase the DPS of low-damage, high-attack-rate weapons against unarmored targets (such as the Katana).
If the player's Strength stat is the most extreme (highest or lowest), Doc Mitchell will either say "Surprised anybody'd want to tangle with you. Heck, you could go Deathclaw hunting with a switch." (for highest) or "That's some serious atrophy, even for someone who's been in bed awhile. It's a wonder you can move at all." (For lowest)
If overweight and slowed, it is possible to drop an item, or a stacked item, that reduce your weight below the overweight status and return movement to normal. You pick up the item (or stacked item i.e 20 cans= 1 item stacked) carrying it in front of you and once at the destined point hit the pickup to add it to your inventory as you should be holding it. Stow it or whatever at that point and continue normally. Note: Fast traveling will make you drop the item and you cannot carry it through doorways. You will need to stow items if you need to fast travel and come back.