"Teddy Bear, Teddy Bear" is a jump-rope rhyme. You mimicked, while jumping rope, what the Teddy Bear did:

Teddy bear, teddy bear, turn around
Teddy bear, teddy bear, touch the ground.
Teddy bear, teddy bear, show your shoe.
Teddy bear, teddy, bear, that will do.
Teddy bear, teddy bear, go upstairs
Teddy bear, teddy bear, say your prayers
Teddy bear, teddy bear, turn out the light
Teddy bear, teddy, bear, say goodnight.

"My Jolly Playmate/My Vicious Enemy" is a clapping game:

My Jolly Playmate
Come out and play with me
And bring your dollies three
Climb up my apple tree
Slide down my rainbow
Into my cellar door
And we'll be jolly friends
For ever more

My jolly playmate
I cannot play with you
My dolly has the flu
Boo hoo hoo hoo hoo hoo (alternately, "She threw up in my shoe")
Don't have no rainbow
Don't have no cellar door
But we'll be jolly friends
For ever more

alternately...

My vicious enemy
Come out and fight with me
And bring your bumblebee
Climb up my poison tree
Slide down my razor blade
Into my dungeon door
And we'll be vicious enemies
For ever more

I'm not sure which is scarier -- that I remember the words or that I still remember all the hand movements.