Sooooo, bottle carbonation...
You need airtight bottles that will hold pressure. These can be soda bottles, beer bottles, sparkling wine bottles -- basically any bottles designed to hold pressure.
You need live yeast. This is probably a given unless you did something to kill the yeast like heat pasteurize.
You need a correct amount of fermentable sugar in the bottle for the yeast to eat. This is called priming sugar. Lots of "priming calculators" are available online to help you target a particular level of carbonation; the carbonation level is measured in volumes of carbon dioxide. My opinion: 1.5-2.2 vol is low, 2.3-2.7 is moderate, 2.8-3.4 is high, 3.5+ is probably excessive (and potentially dangerous).
***Too much sugar (or uneven mixing) may result in gushing bottles or explosions from the pressure.***
This is why we need a controlled amount of sugar, and it needs to be fully dissolved and mixed in the batch being bottled.
The yeast needs time to ferment the sugar. This may take days to weeks.
Temperature has a large impact on the speed the yeast create carbonation. You want to store the bottles at the same temperature where the batch was fermented, at minimum.
Refrigerated bottles will probably not become carbonated in a reasonable amount of time.
Hope this helps.