Dashing through the snow
In a one-horse open sleigh

O'er the fields we go
Laughing all the way

Bells on bobtails ring
Making spirits bright

Horses often have their tails "bobbed". This generally means that the horse's tail is folded up and tied (like an old lady's bun) to make it look shorter and get it out of the way. If you put bells on a horse's bobbed tail, they will ring as the horse moves.

A day or two ago
I thought I'd take a ride

And soon, Miss Fanny Bright
Was seated by my side

The horse was lean and lank
Misfortune seemed his lot

He got into a drifted bank
And then we got upsot!

A gent was riding by
In a one-horse open sleigh
He laughed as there I sprawling lie
But quickly drove away

Now the ground is white

Just get a bobtailed bay
Two forty as his speed
Hitch him to an open sleigh
And crack, you'll take the lead