Difference between "Sweet" , "Yellow" ,"White" and "Chives"
There's a great guide to onions that covers most of these and more over on Serious Eats (all of the quotes below are from this article). But, in relation to your question:

White onions are the standard onion you'll see. They are white.



Serious eats describes them as:


  The white versions are somewhat sweeter and cleaner in flavor, but don't store quite as well as yellow onions do. [. . .] Milder in flavor than yellow onions, white onions can be eaten raw.


Yellow onions are apparently the most popular type of onion in the US. They are yellow but otherwise pretty similar in appearance to white onions.




  Yellow onions are undoubtedly Americans' favorite: nearly 90 percent of onions grown in the US are yellow. Their deep but not-too-strong flavor makes them endlessly versatile in cooking. Larger, slightly sweeter yellow onions labeled Spanish onions are often found right next to plain old yellow onions; they're a milder choice that works well for raw applications.


Sweet onions are specialty onions... there are a bunch of varieties. In Texas, you're probably familiar with the "Texas 1015". Similar varieties are the Vidalia of Georgia and the Walla Walla of Washington.



These onions are yellow like a yellow onion but they are more saucer-shaped than round. They're very sweet and can be eaten raw pretty comfortably.


  Super-sweet and crisp, ideal for eating raw.


Your "sweet onions" are probably what the Serious Eats description classifies as "young onions" they fit your description - visually similar to green onions/scallions (they are the same thing) but with a more bulbous base.




  Though spring onions resemble scallions in appearance and flavor, they're actually just very young storage onions—yellow, red and white—that are pulled out of the ground at an earlier date, when they're still thin-skinned and mild in flavor.


As to your question about chives, they resemble tiny scallions. They're sold in bunches but aren't really layered the way a scallion is.



They're generally sold in bunches and don't have white bulbs at all. They're generally just used as a topping, like on baked potatoes. They have a pretty mild onion flavor.

Here's a fun infographic from Buzzfeed about different types of onions: