Fortis in Latin means strong, firm. Lenis means slow, calm, soft.
When the air escapes through an open glottis (for a voiceless consonant), it's not slowed down by anything, and arrives at the stricture at full strength. When the vocal folds are vibrating, they slow the airflow down, and therefore the pressure before the stricture is lower.
The stricture has to be more firm for voiceless consonants, in order to sustain the higher pressure.
Metaphor story by Dr.Machač, slight improvements by myself:
Imagine Frank Fortis and Luke Lenis want to play football (soccer, for Yankees :-)). They need the goals. Frank is punctual, and wants a firm goal which can sustain a fast ball. He takes many bars, nails, and ropes and creates a very stabile goal. It takes time, but it holds. Luke, on the other hand, is lazy. He just takes two bars, plucks them into ground and puts another across them. It takes just a few moments to build, but a faster ball would break it. But that doesn't matter, because his partner is weak and Luke can expect only slow shots.
Unfortunately, their neighbour, Mr.Sam Succeeding Sound doesn't like their play and wants them to dismantle their goals. Luke just pulls out the bars and he can go, but the poor old thing Frank has to disassemble his complex construction, and that will take time. It was worth the effort, though, because Luke is strong and his shots were fast, but Frank's goal sustained it.
There is a wide open window in the room. You need the door to be half-open (half-closed, for pessimists). You have to put a lot of effort, because the draft is so strong it would flung the door open. The window is the vocal folds, the door is the stricture, and you have to be fortis.
However, if you fasten the window so that the air can go only through a little cranny, you won't need so much effort to keep the door where you want them to be.
In some languages, where phonologically voiced consonants are really voiced from the beginning to the end, like e.g. Czech or Spanish, this is not so important for the perception. It would be just an interesting piece of trivia. It only allows us to understand the whispered speech. Try to whisper for example the Spanish "panal" and "banal", and ask somebody if s/he understands. For Czech, the typical example is "saze" and "zase". Notice that there's a big difference between a "s" and a fully devoiced "z".
Unfortunately, some languages, including English, German, or Japanese, the vocal folds don't necesarilly have to be vibrating during a phonologically voiced consonant. We call it a consonant partially or fully devoiced. Here, it's not so much important whether the vocal folds are actually vibrating or not, rather whether the pronunciation is fortis or lenis. Of course, to give contrast between phonologically devoiced and voiceless consonants, these languages often use aspiration.
We don't speak about fortis or lenis in some consonant types:
Nazals: There's a very small rising of pressure before the stricture, as all the air escapes through the nose. According to our metaphor, the draft doesn't affect the door you're holding, as it escapes through another, wide open door next to yours.
Approximants: The stricture is so open that there's a very small rise of pressure before it. Metaphorically, you don't need the door to be half open, but nearly open (not entirely, though).
Taps, flaps, trills: You need the draft to be strong enough to rattle the door.