Skyrim:Carry Weight

Carry Weight is the maximum weight of all weapons, armor, and other items that you can carry. It is the equivalent of the Encumbrance attribute in Oblivion. All characters start with a Carry Weight of 300.  If you try to carry a combined weight that is greater than your Carry Weight, you will be limited to walking, unable to run or sprint. 

You will also be unable to fast travel while over-encumbered unless on a horse.

Carry Weight is only controlled by your base Stamina, and therefore temporary changes to Stamina, such as Fortify Stamina, do not have any effect on Carry Weight; this is also why you do not become over-encumbered every time you swing a weapon or do other activities that deplete your Stamina.

Current Carry Weight

You may inspect current carry weight in the bottom right of Inventory menu.

Increasing Carry Weight

# Choosing to increase Stamina when leveling up increases your maximum Carry Weight by five points.
# Carry Weight can be increased by 100 through the Extra Pockets perk in the Pickpocket tree.
# The spell effect provided by the Steed Stone also adds 100 points to your total, as well as completely negating the weight of any equipped armor.

Circumventing Over-Encumbrance
Moving faster. You can walk at a faster pace while over-encumbered if you repeatedly do forward power attacks and sheathe right before the stamina drain, ideally with a dagger. The Whirlwind Sprint shout can also aid moving forward, especially if you unsheathe a weapon during the sprint, which causes you to run a vast distance at incredible speed.
You can also walk at a slightly faster pace if you have a bow fully drawn while moving forward.
Potions. You can drink a Potion of Strength (or a better quality version), and fast travel before it wears off, provided the weight you are carrying does not exceed your increased carrying capacity.
Enchantments. You can carry gloves, boots, and/or pieces of jewelry enchanted with Fortify Carry Weight, and equip them when you exceed your limit in order to fast travel. Clothing such as the zero-weight shoes are particularly useful for this.
Transform. Werewolves using their beast form have their carry weight limit increased by 2000 points.
Transforming into a Vampire Lord  will also allow you to avoid being over-encumbered and travel faster.
Horses. Horses will take you where you want to go regardless of over-encumbrance. You can also fast travel after mounting a horse. Arvak  is especially handy for this, as he can be summoned when you need him, then released when you don't.
Eat some food, especially unwanted.
Drop and drag. Drop a heavy item to remove over-encumbrance, and hold the Activate key to drag it while you move (be cautious when loading a save, as this will move the item back to where you've dropped it).
Depositing items in a small dead body, such as a Skeever, can be particularly useful, as it can hold any number of items.
You can dump unlimited amount of heavy equipment into a corpse and reanimate it using a Spell, Staff or power, such as the Resurrection power gained by using The Ritual Stone. The Dragonborn can then fast travel to any destination and the corpse follows. If the time it took is enough for the spell to run its course, the corpse will die again upon reaching the destination, and the Dragonborn can loot the body for all the equipment they gave it initially.
If the Dragonborn lacks a reanimation ability, a corpse can be filled and then flung ahead by holding activate to lift it, then dual-casting Lightning Bolt. Care should be taken not to fling it into water.
Drinking a "Draught of Strength" or any other type of fortify carry weight potion and fast travelling before it wears off is a viable option, as it temporarily increases the carry weight capacity. This is similar to the second tip in this section, in that it allows fast travel to start, but once the destination is reached, the potion will have worn off.
If one has a follower, one can drop an item on the floor, and command a them to pick it up. This seems to negate a their weight limit, allowing them to carry an apparently infinite amount of items. Keep in mind that this method will mean one is unable to actually give an item to said follower, however.
The Dremora Butler ability acquired as a reward for completing "Black Book: Untold Legends" will allow the player to transfer a total of 248 base carrying weight to a temporary servant for 15 seconds.
Alternatively, the player can purchase a house and store items in any container without fear of said items disappearing. The items will stay in the chest until taken back out.
The Abandoned House in Markarth is free to use, being a place with permanently safe containers, regardless of whether the Dragonborn is Thane or not.
Alternatively, Anise and Drela both have cabins that can serve as permanently safe storage places, if the player elects not to purchase any houses.
Another way to overcome encumbrance is to place excessive items on a dead body before raising it as a Dead Thrall. This will allow it to be used as a "pack mule" to temporarily carry extra items.
Theoretically, a Dead Thrall can also benefit from the Fortify Carry Weight enchantment, but this only works if the armor they were wearing was enchanted. In other words, it is possible to remove the corpse's armor to enchant it, then replace it on the dead body. It will not be possible to enchant armor which the character did not wear in life and have them wear it. Corpses which were equipped with more clothing slots when they were alive will fully maximize the value of this enchantment, because the more armor they wore in life, the higher the carry weight they can benefit from.
Using console commands.
tgm" – (Toggle God Mode) will enable running, sprinting, and fast-traveling while encumbered, but will not remove limitations on the use of carriages.
player.ModAV Carryweight   – will permanently increase your carrying capacity by "#" points.
Followers and Spouses can be used to carry extra items, although they too have their own limit that cannot be exceeded. (Total carrying capacity is approximately 290 units of weight)
However, it is possible to use the glitch in Multiple Followers to gain 3 to 4 followers, thus effectively multiplying the player's total capacity to 1,000 units of weight. This can be performed using Cicero and Dark Brotherhood Initiates, along with a fourth companion.
Followers can also benefit from the Fortify Carry Weight enchantment, although this is bugged. Should their original (unfortified) limit be exceeded, only one item or stack of items can be added, then the menu must be closed and reopened to add more, up to the absolute maximum capacity.

Notes
You cannot execute a sneaking forward roll (ordinarily usable with the Silent Roll perk) when over-encumbered.
Carry Weight enchantments (even multiple) work on followers, but with a bug: As soon as you reach the normal (unfortified) carry weight limit, every single item you put on your follower makes the inventory menu react as if the carry weight limit would have been exceeded. To become able to add one (!) more item, you have to leave the dialog and reenter it, for every single item up to the real (fortified) carry weight limit. After reaching the latter, toggling the inventory menu no longer opens space for more items.

Bugs