Strength measures a characters ability to exert physical force. A character with high Strength can lift heavier objects, carry more gear without being overloaded, break things with brute force, shove and grapple creatures more effectively, and is more accurate and more effective with melee weapons.
This is most important for characters who fight in melee using heavy weapons like longswords and greataxes. Barbarians, Fighters, and Paladins frequently have high Strength.
Athletics is the only skill tied to Strength.
Athletics covers a large number of physically strenuous activities, including climbing, jumping, and swimming. While you can typically perform these activities without making an ability check, the DM may call for an ability check when the circumstances make the activity difficult. For example: You don’t typically need a Strength (Athletics) check to climb stairs, but you might need one if you are climbing a sheer cliff. Similarly, you don’t need a Strength (Athletics) check to swim across a still, shallow pool, but you might need to make a check if you are attempting to swim across a rushing river.
Athletics is also used for shoving and grappling, both of which will be covered in the Combat section, later in this guide.
Lifting, Carrying, and Encumbrance
Strength also dictates how much a creature can lift, drag, and carry. The amount of weight that you can handle is a multiple of your ability score, which is one of the very rare cases where your actual ability score is used instead of your modifier.
You can carry 15 times your score. This capacity allows you to handle any clothes, weapons, armor, or other items that your character has on their person. This number is high enough that you usually don’t need to worry about it, even if you character’s Strength is poor, unless you’re doing something silly like trying to carry a horse around. Most groups simply ignore the rules for carrying capacity (except in cases of obvious abuse), but if it becomes a problem you can always buy a mule.
You can drag, push, and lift 30 times your score. In this case “lift” means what you can pick up off the ground, but you probably can’t carry it very far. It’s a bit like going to the gym and lifting the heaviest thing you can manage. Sure, you may be able to lift something very heavy, but you’re not going to take it for a walk around the neighborhood.
Creatures of Small and Medium size use the same rules for carrying capacity, lifting, etc. This means that a halfling with 10 Strength can lift 300 pounds despite weighing roughly 30 pounds. No, it doesn’t make any sense, but a lot of realism is traded away in favor of making the game playable.