Learn Farsi: گلدانی کنار تلویزیون است.

In Farsi, spatial words like kenâr (next to) need an Ezafe (an "e" sound) to connect them to the noun they are pointing to. So, it is pronounced kenâr-e televizyon. However, the Ezafe is a short vowel, and short vowels are almost never written in the Farsi alphabet. You just have to know to pronounce it when you see a spatial preposition followed by a noun!

What is the letter added to the end of goldân?

That is the indefinite suffix -i. Adding it to the end of goldân (vase) changes it from "vase" to "a vase" (goldâni). Because Farsi letters connect to each other, the letter Ye is simply attached to the end of the word.

Why is the verb ast at the very end of the sentence?

Farsi naturally uses a Subject-Object-Verb word order. Even in simple sentences describing where something is, the verb for "is" (ast) waits patiently at the very end. Literally, this sentence is structured as "A vase next to the television is."

Could I change the word order to put the location first?

Yes, you could say kenâr-e televizyon goldâni ast. However, Farsi uses word order to show emphasis. Putting the subject (goldâni) first is the most standard, neutral way to state this fact, whereas putting the location first emphasizes the area next to the TV.