If it is extremely crowded, you can say "packed like sardines in a can" or just "packed like sardines" or "packed." This comes from the way sardines are tightly packed into cans when canned for eating:

Strictly speaking, some people will object to this usage: both the place and the people in it can be said to be packed, but only the people can strictly be packed like sardines since the sardines are inside the can and the people are inside the crowded place. However, colloquially people will use the phrase in both senses. Also, most people just say packed:

The subway wasn't working so the buses were all packed.

There was no room to swing a cat? We were pressed together like peas in a pod (often used to indicate uniformity, but sometimes literal closeness)? You couldn't fit a knife/cigarette paper between the [contestants]?

Adding to the nice suggestions so far, I've found one dictionary that offered "there is hardly breathing space".

sufficient space in which to move, work, etc.: The train was so crowded that there was hardly breathing space.

In British slang, we often describe a place as "rammed", when full:

(be rammed) British informal (of a place) be very crowded:
    the club is rammed to the rafters every week

Packed to the gills

Packed to the rafters

Packed to the gunnels (from gunwales <- gun walls) (originally of a ship)

A popular East Side bar, packed to the gunwales with arch young bankers and ersatz Now girls.

Pressed like olives/grapes

packed together tighter than two coats of paint

Bursting at the seams (the room was...)

As crowded as a beehive

Big and poor and as crowded as a beehive.

And since we are photo-heavy...

"I would rather sit on a pumpkin and have it all to myself, than be crowded on a velvet cushion." - Henry David Thoreau

"Like Picadilly Circus" is commonly used in en-GB.

For a more international audience, I'd go with "packed like sardines".

"It's like the Black Hole of Calcutta in here." This refers to the tiny dungeon in which British prisoners of war were held in June 1756 after the fall of Fort William, Calcutta, where (allegedly) the vast majority perished from suffocation or heat exhaustion because so many people were crammed into such a small space.

For a one word equivalent of "crowded": crammed, rammed, heaving, packed.

One venerable — nay, agèd — word for overcrowded is opplete, which is both an adjective and a verb.

Quoth the OED regarding the adjective:

oˈpplete, ppl. a. Med. Obs. Also 6 opplet.

Etymology: ad. L. opplēt-us, pa. pple. of opplēre: see next.

Filled up, crowded.

• 1545 Raynold Byrth Mankynde P j, - The coti lidons be opplete, stopped, & stuffed with yll humours.
• 1578 Banister Hist. Man i. 34 - The posteriour part [of the leg] is opplet, and filled with much store of flesh.
• 1646 J. Hall Horæ Vac. 134 - How should they not be opplete with grosse humours?

Quoth the OED regarding the verb:

oˈpplete, v. Med. Obs. rare.

Etymology: f. ppl. stem of L. opplēre to fill up, f. ob- (ob- ^{(def#1) (def#d)}) + plēre to f ill.

trans. To fill up, fill to repletion.

• 1620 Venner Via Recta ii. 41 - They..opplete the ir bodies with waterish, crude, and windy humors.
• 1620 Venner Via Recta iv. 71 - That it be not..oppleted with much fat.

More modern formulations for a crowded place include the idiom that a place is teeming with people. The OED says that teeming as an adjective means

Abounding; swarming; crowded.

Another word you might be looking for is the noun throng. The OED includes these two relevant senses for it:

2. Pressing or crowding of people; an act of thronging or crowding; crowded condition.

3. concr. A crowded mass of persons actually (or in idea) assembled together; a crowd.

There is also a verb throng to accompany the noun, whence derives the adjective thronging meaning pretty much the same thing as teeming. Per the OED:

The action of the verb throng; pressing; crowding.

How about "chock-full"?

adjective
[predic.] informal
    filled to overflowing:
        my case is chock-full of notes

And the British term "chock-a-block".

adjective
[predic.] informal, chiefly British
    crammed full of people or things:

• 'A sea of people.'
• 'A writhing sea of humanity.'

Let's not forget the conventional and all-purpose full house.

full house

noun

an audience, or a group of people attending a meeting, that fills the venue for the event to capacity.

"Assholes and elbows"

In a crowded place (such as a bar or night club), it's often hard to maneuver for all the assholes and elbows filling up the place.

Packed is the easiest word that comes to mind.

packed

noun

• cram a large number of things into (a container or space).
  • "it was a large room, packed with beds jammed side by side"
• (of a large number of people) crowd into and fill (a room, building, or place).
  • "the waiting room was packed"

The place was so crowded that you couldn't swing a cat. The room was full to the extent that there is no elbow room. The street/square was packed to the rafters. A crowded event might be described as standing-room only.

Irish slang - possibly moribund by now: "jammers" as in - the bar was "jammers"

"It's like Grand Central Station ..."

Referring to the hustle and bustle of the very famous and busy railroad terminal in New York City.

The place was chocker.

This basically means the place is full, and is presumably an abbreviation of chock-a-block.

You could also say the place was heaving.

How about, "It was a mob scene."

Or perhaps "an ugly mob scene," if it was anything like our local shopping malls this past holiday season...

"That was a clusterfuck."

The first thing to come to my mind when I read the question.

E.g. Urban Dictionary

A corpus search (Google for clusterfuck and crowded) yields 1.2 million results.

A local, possibly New England, expression is "stuffed tighter than a turkey" (or other colorful variation).

Also, as this Houston Chronicle article demonstrates, packed tighter than a pair of Wranglers (or designer jeans) is a relatively common expression.

From Oxford Dictionaries Online:

brimming

adjective

• fill or be full to the point of overflowing. "a brimming cup"

and

overflowing

adjective

• (esp. of a liquid) flow over the brim of a receptacle.

and

overcrowded

adjective

• filled (accommodations or a space) beyond what is usual or comfortable.

This might work ... how about 'jam-packed' ?

http://www.thefreedictionary.com/jam-packed

The place was so crowded you had to go outside to change your mind.

This quip is often used when talking about the size of a cramped room or house, as one writer mentioned in an article about language sometimes used by real estate agents:

For example, he cites the commonly used term "cozy" and says the connotation to savvy Realtors is that there isn't much space in the house.

“It triggers the Henny Youngman in us: ‘This house is so small that you have to go outside to change your mind,’” Boyd says.

Bazaar It is urdu for market . It is suitable as asian markets are usually crowded and noisy.

Irish has the expression dubh le daoine, and a literal translation of that — “black with people” — is used in Irish English to describe a crowded space. According to a post on boards.ie, this can sometimes be abbreviated to “it’s black in here” to describe a crowded space.

(Interestingly, I learn also from boards.ie of an equivalent expression in French: être noir de monde.)

Infestation. There's no reason to elevate one species above others.

Depressingly, the sole local discotheque was packed like the hold of a slave ship. People were crammed in there like the black hole of Calcutta.