First learn the terms which define typography variations. (Here's a decent link for that - and another- or Google search for "basic typography terminology") In order to actually identify any typeface, you need to understand exactly what defines that type face... its serifs or lack there of, its counters, its descenders, ascenders, tails, terminals, kerning, bowls, etc. Knowing what to look for is half the battle, if not more. This is how people can identify any typeface - by a unique aspect (most often found in characters such as lowercase a, g, q, etc.)
From there it's really more a matter of usage and familiarity. The same way you may recognize a quote from your favorite movie, because you've seen it 50 times, if you use a typeface a great deal, you will become familiar with that typeface's serifs, descenders, ascenders, counters, tails, etc. now that you know what to look for. No one learns exactly what makes Frutiger Frutiger. They learn what serifs look like and then after seeing a unique serif on a typeface, then identify the typeface as Frutiger and therefore learn that Frutiger has those serifs - making it possible to immediately know what Frutiger looks like by just looking at a serif.
In terms of application, or when to use what within a design, you need to understand the psychology which is conveyed by certain aspects of typography. Things such as "serifs appear more friendly and make readability easier in print" and "serifs can make readability harder in smaller sizes on the web." Or... "small typography with large counters increases legibility" or.. "bold sans-serif type tends to convey a sense of being 'official' or 'institutionalism'." (Here's a decent infographic regarding the psychology of a typeface -- it's subjective, like color psychology, so I may not absolutely agree with everything here, but I can't state anything is "wrong" either.)
As far as someone telling you when to use a specific typeface or what specifically would "look good" there --- that's all just opinion and then a reference to some typeface they are familiar with. There are no rules where this is concerned. Many designers have a stable of 5-10 typefaces they tend to use regularly. So, they are familiar with those typefaces and will often suggest what they are familiar with. That doesn't mean they are correct. (I'm not casting any aspersion here -- I do this as well.) For example, when I hear someone suggest Open Sans for a printed piece... I know they are familiar with web fonts, because Open Sans is wonderful on a web site (like GD.SE) because it was designed for web usage, but it often fails in print design in my opinion.
With all the above you can quickly determine that based on what you wish to achieve or convey in a design using a sans-serif font may be best, so you can ignore all typefaces which are serifs or scripts. Or you may decide you need something highly legible at a very small size, so you'll want to use a serif typeface with larger counters.
This is all done similar to how if you are creating something you want to appear "professional, corporate, secure, and dependable" there may be a tendency to instinctively know (or "feel") that using fluorescent, loud, vibrant, colors would be a contradiction to the message you are trying to convey. After familiarity with the psychology behind typography there's the same tendency to instinctively know that using a script or display face may be an equally poor choice. Or if creating a piece for a child's toy... the exact opposite of these choices may be the best path to pursue.
Repeat... Most designers have a collection of 5-10 fonts they use regularly and are familiar with. They tend to stick to these typefaces for much of what they create. Any deviation is intentional.
There's no such thing as "a font that says fun" or "a font that looks professional". Fonts don't generally convey anything in themselves - Helvetica looks as "fun and professional" as Palatino or Optima do. It's the usage of any typeface which can do more to convey a message.
Think of it like having 10 identical outfits.. you never need to think about what you're going to wear on any given day. But... if it's colder outside you may need to deviate and wear a sweater too. Or if it's warm, deviate and wear shorts instead of pants. But the base decision has been made and there's no need to think about it until external forces cause a deviation.
So, start by picking out 5 general typefaces... a couple serifs, a couple sans-serifs.. a script... and perhaps a "specialty" font. If you don't have one, getting a Font Manager application can help considerably. You can deactivate all the fonts you aren't using.
Then a design starts with these defaults... and can they change typefaces if one of the "defaults" is problematic. If you run into a problem you can then seek to replace that "default" choice.. so you'll know you need to examine other serifs, or san-serifs or scripts, etc. With a font manager, you can filter lists or see typefaces before activating them.
I generally start every design with 2 fonts (a serif and a sans-serif I like)... then will alter typefaces only when necessary. Example: Older audience so I need a serif with larger counters. Or need a third choice for a subhead line font, a semi-serif perhaps.
Massimo Vignelli only used Helvetica. That's all - he felt he never needed anything else.