At a first sight, one of these seems to be wrong. It's either "jewelry" or "jewellery", but one of them is certainly a mistake, right? Well, no. But let's take a closer look and understand why both forms are accepted and when you should choose the right spelling according to your context, so that you master the English vocabulary and use it with elegance.
Jewelry vs. Jewellery
So we all know what a jewel is. And it is also easy to correlate its meaning with "jewelry" or "jewellery" so we know that they both refer to the same concept. Therefore, both "jewelry" and "jewellery" refer to objects and accessories such as earrings, bracelets, necklaces, studs and rings, made out of gold, silver and other precious stones, worn as decorations. Then what is, after all, the difference between "jewelry" and "jewellery"? Why do we spell them differently?
The answer is simpler than you might expect: it's the variations between British and American English. It's "jewellery" in UK and "jewelry" in the US. Pretty much clearer now, isn't it? Well, that's all you have to know about this pair of words.
When do we use "jewelry"?
Thanks to the fact that it's shorter, "jewelry" is a lot more often used today. After all, it's more comfortable to type or spell "jewelry" than the longer form. Even so, it won't be considered a mistake if you choose to spell "jewellery", given the fact that your message will still stay the same and you'll still refer to the same precious accessories. Spelling it like this is a simple matter of choice. But of course, it is more elegant and recommended that you use "jewelry" especially in a conversation in US English.
When do we use "jewellery"?
Just like in the previous case, "jewellery" is certainly recommended if you are communicating in British English. It's a matter of consideration for your interlocutor's preferences and culture. Even so, if you forget this small detail or simply prefer to spell it shorter, the word won't be misspelled, according to all dictionaries. It is just more elegant and formal if you choose "jewellery" in UK English and keep the "jewelry" spelling for US English.
Both "jewelry" and "jewellery" are correct and you can replace them anytime with no restriction regarding their meanings. It is, anyway, about the subtle differences between UK and US English and so it's recommended that you use the first in American English and the second in British English.