Mystery:
According to the dictionary:
- anything kept secret, unexplained or unknown;
- a person or thing that arouses curiosity or wonder;
- the quality of being hidden, hard to understand, or puzzling:
When we talk about mystery fiction, it usually means the plot is focused on solving a problem, typically a crime. However, the problem may be simply an unexplained phenomenon, not a crime and maybe not even a problem, just an event that piques one's (reader and character's, hopefully) curiosity.
As for mystery within the story, just look at the dictionary definitions. Any plot or subplot point that pique's one's curiosity and requires some effort to understand. It can be a character's hidden backstory, a locked room in a house, a family event that everyone refuses to acknowledge, a downpour over the campfire while the rest of the field is dry (that's a real metereological phenomenon, by the way, but being both a wonder unknown to most people and puzzling even for those who study weather, it deserves to be counted as a mystery).
Suspense:
According to the dictionary:
- a state of mental anxiety and uncertainty, as in awaiting a decision or outcome.
Basically, it is an effect produced with a good plot and the right techniques. Mystery may help (in the sense of something unknown and unexplained) but only if there's a sense of danger related to it. Maybe a woman's newest boyfriend avoids talking about his past but, unless he has a fascination about knives, guns and excusing psychos, there's little reason to feel anxious. Unless you're the woman's parents, obviously, but that's 'parental anxiety' and has little bearing here.
Tension:
According to the dictionary:
- emotional strain, esp. intense suspense, anxiety, or nervousness;
- a strained relationship between individuals, etc.
At first sight, seems almost the same as suspense, right?
Suspense and Tension:
This is mostly an empirical answer based on my personal experience as a writer, reader and student of literature (unfortunately, the classes didn't cover this particular topic).
Suspense must have tension, but tension doesn't necessarily imply suspense.
A strained relationship will have tension (whether it's sexual or not). There is an emotional distress or struggle, for example, there is a deep longing for something/someone that one doesn't know how to obtain so they anxiously hunt (or wait) for it. If the writer does their job right, the tale will capture the reader's attention and will sail the growing tension, eager to see where it will go. There's pleasure surfing that wave of growing tension and little fear; it's a leisurely read.
Still, it won't be suspense until a notion of danger sips in. Maybe the relationship is leading someone into the grip of a murderous stalker or maybe one of the partners risks losing job, house, even family. Now the reader is surfing a mixed sea of pleasure and fear: will it end in tragedy or in a happy ending? The reader doesn't just want to see where it's going, the reader wants to know what is going to happen next, and they want it NOW. To put down the book is to prolong the agony of not knowing who will die (literally or figuratively).
EDIT:
It seems I just keep over-looking second questions...
there can be a tension -in terms of emotions- even without a particular character, no?
If the readers don't care for the character who is under stressful situations, they'll be less likely to feel the tension as their own.
On the other hand, I can personally get gripped by suspense in a story even if I dislike the character. Of course I'm one of those who, if you can't guess how it'll end, then I'll want to know. And even if I hate a character, if everything else is well created, I'll be biting my fingernails to the bitter (or sweet) end.
For me, it's more about the plot capturing the reader rather than the character. Furthermore, if the plot is poorly crafted, there's no sympathy for the character that will make me carry on.