Many tongue-twisters use a combination of alliteration and rhyme. They have two or three sequences of sounds, then the same sequences of sounds with some sounds exchanged. One example is Betty Botter 

Betty Botter bought a bit of butter.
The butter Betty Botter bought was a bit bitter
And made her batter bitter.
But a bit of better butter
Makes batter better.
So Betty Botter bought a bit of better butter,
Making Betty Botter's bitter batter better.

Some tongue-twisters are short words or phrases, which become tongue-twisters when repeated rapidly, often expressed as "Say this ten times fast!".

"Wet rain."
"Peggy Babcock"
"Thin Thing"
"French friend"
"Red Leather, yellow Leather"
"Unique New York"
"Sometimes sunshine"
"Irish wristwatch"
"Toy Boat."
"Flammable Flanimal"
"Big whip" is another that is difficult for some people to say quickly, due to the lip movement required between the "g" and "wh" sounds.

Some well-known English tongue-twisters are:

Billy blew a blue bubble while bouncing on a bongo.

"Peter Piper":

Peter Piper picked a peck of pickled peppers,
A peck of pickled peppers Peter Piper picked.
If Peter Piper picked a peck of pickled peppers,
How many pickled peppers did Peter Piper pick?

But if Peter Piper picked a peck of pickled peppers,
Were they pickled when he picked them from the vine?
Or was Peter Piper pickled when he picked the pickled peppers
Peppers picked from the pickled pepper vine?

"How much wood would a woodchuck chuck?":

How much wood would a woodchuck chuck
if a woodchuck could chuck wood?
A woodchuck would chuck all the wood that he could
if a woodchuck could chuck wood.

A woodchuck
would chuck
all the wood he could
if a woodchuck could chuck wood.

"Sick hicks":

Six sick hicks nick six slick bricks with picks and sticks.

"Sister Suzie":

Sister Suzie sewing shirts for soldiers
Such skill at sewing shirts
Our shy young sister Suzie shows
Some soldiers send epistles
Say they'd rather sleep in thistles
Than the saucy, soft short shirts for soldiers Sister Suzie sews

"She sells sea shells":

Sister Sue sells sea shells.
She sells sea shells on shore.
The shells she sells.
Are sea shells she sees.
Sure she sees shells she sells. 

"Night light":

You've known me to light a night light
on a light night like tonight.

or a variation:

There's no need to light a night light
on a light night like tonight,
for a night light's a slight light
an a night like tonight.

"Thistle sifter":

Theophilus Thistle, the successful thistle sifter,
in sifting thousands of unsifted thistles,
thrust thrice three thousand thistles
through the thick of his thumb.

"Jello":

Orange jello, lemon jello,
Orange jello, lemon jello,
Orange jello, lemon jello.

This next one won grand prize in a contest in Games Magazine in 1979^]: (Contest announced in issue of November/December 1979; results announced in issue of March/April 1980).

Shep Schwab shopped at Scott's Schnapps shop;
One shot of Scott's Schnapps stopped Schwab's watch.

Spoonerisms

Some tongue-twisters are specifically designed to cause the inadvertent pronunciation of a swearword if the speaker stumbles verbally (see spoonerism).

An English example of this sort:

I'm not the pheasant plucker, I'm the pheasant plucker's mate,
And I'm only plucking pheasants 'cause the pheasant plucker's late.
I'm not the pheasant plucker, I'm the pheasant plucker's son,
And I'm only plucking pheasants till the pheasant pluckers come.

Or another:

I'm a sheet slitter
If sheets need slitting,
Sheets I slit.

Or another:

I'm a sock cutter and I cut socks.

And Another

Fuzzy Wuzzy
Fuzzy Wuzzy was a bear
Fuzzy Wuzzy had no hair
Fuzzy Wuzzy wasn't fuzzy, was he?