What is meant by "neutral" oils? Conversely, what would make an oil not neutral? When is it important that the oil be neutral (or not)?
Context would make me more sure, but I expect this is referring to neutral flavor.
If you need oil in something for purposes other than flavor, and want to make sure that the other flavors in the dish aren't affected or obscured, then you'd want a neutral oil. For example, for frying and sauteeing, we rarely use strongly flavored oil. (There's the additional problem that many flavors are destroyed at high heat; many more aromatic oils have lower smoke points.)
Oils like sesame oil, chili oil, and walnut oil are decidedly not neutral. On the other end, oils like canola oil and grapeseed oil are neutral. Close to neutral, there are very mildly-flavored oils like peanut oil. Somewhere more in the middle there are things like olive oil, which has some flavor but isn't overwhelmingly strong.
Note that sometimes there's more than one type of oil from a given source, often a more refined one that's more neutral and a less refined one with more flavor. For example, refined olive oil is quite neutral, but extra-virgin olive oil has plenty of flavor; "regular" peanut oil is fairly refined and mild, but there are also flavorful roasted peanut oils.
A neutral oil simply refers to the flavor. Olive oil, for example, is not neutral, it brings a distinct range of flavors to a dish. Grapeseed oil is considered one of the most neutral oils. It is nearly flavorless and can be used for frying, making herb oils, and other situations where you may not want any noticeable flavor from the oil medium.
Canola oil is definitely not neutral! To me, it has a distinct strong flavour that would spoil many dishes. This is most probably a cultural distinction. What's neutral for you may not be neutral for me. I'd consider sunflower oil neutral, or even non extra virgin olive oil (second press). Mayonnaise with canola oil tastes very synthetic. Perhaps in countries with poor quality refined oils, people may think that refined oils seem neutral when they actually have a strong flavour. I suspect a cultural influence here.