"Koń chodzi."

Translation: A horse walks.

[JamesTWils]

Are chodzi and ide (I cannot remember if that is the proper 3rd person singular) completely synonymous in Polish?

[jmango2]

No. Polish verbs of motion, like in other Slavic languages, distinguishes between determinate and indeterminate verbs.

What that means is that chodzi refers to frequent trips and movement in general. Idzie on the other hand, refers to a a single trip going somewhere specific.

Verbs of motion are one of the bigger points of trouble for native speakers of Germanic and Romance languages, so I'd suggest Googling "Polish verbs of motion" to find out more.
This site has a pretty good explanation, but it's gonna take a while to get used to: http://www.polish-dictionary.com/polish-motion-verbs

[lup0]

I put the horse is going. Is that not right here?

[immery]

In different contexts
The horse is walking/ The horse goes.

could be correct.

chodzi = has ability to walk / is walking/walks aimlessly/ goes/walks (somewhere) regularly

This particular short sentence seems to me as is walking aimlessly or has ability to walk

[LordAlex22]

Is the knight chess piece also called "koń" in Polish? Can I say: "Koń chodzi po literze L"?

[Vengir]

Colloquially some people call it „koń”, but the proper term is „skoczek” (lit. "jumper"). And you wouldn't say „chodzi” (because it has no legs), but rather „porusza się”. So in short: „Skoczek porusza się po literze L”.