Roughly, to most people, a "strong" deck is one that'll tend to win against other decks in its format, and a "weak" deck is one that'll tend to lose. It's about winning and losing, though, not as much how you win. Note that "in its format" is a pretty big deal - strong Standard decks are completely different from strong Vintage decks.
Why isn't speed of winning a good way to look at this? Well, not all decks with the same chance of winning do so at the same speed. Especially in Standard, it's normal for there to be aggro decks, midrange decks, and control decks that are all reasonably strong (they all have a good chance of winning a lot facing diverse opponents' decks). The aggro decks can win fast, the midrange decks take a bit of time, and the control decks always take a while. But people will quite reasonably say there's a strong deck in each of those categories, and if they try to compare, it won't be based on speed (then the aggro deck would always be "best"), it'll be based on which they think they can win the most.
If you want to try to measure, I'm afraid playtesting is really the only way. You have to try playing the decks against all the other decks you think might be strong (or find others who have done so), and play plenty of matches. This is why there's so much debate about what decks are strongest: it's difficult and time-consuming to measure, so people try to shortcut by testing just a bit and then using their experience and knowledge to fill in the gaps.
Once you look more closely, a few things make those statements a bit difficult to evaluate and measure.
First of all, people's subjective preferences do come into play, and on top of that, they'll tend to exaggerate, or at least be a bit cavalier with claims. People can also have objective preferences; they might be better at playing with some types of decks than others, and thus consider them stronger.
Next, "tend to win" might not indicate as large of a probability as you'd think. For example, someone might reasonably say "that deck is really strong" when it's going to win 55% of its games against other decks. That may not really sound that strong, but on the other hand, if a decent number of people play the deck at a tournament, that makes it likely that a large fraction of the top N players of the tournament will be ones playing that deck.
So if people are talking about "strong" and "weak" in the context of a tournament (or serious competitive play in general), the difference between the two might be relatively small, and hard to discern unless you're a fairly skilled player and have played plenty of games with the decks in question (or otherwise have data). If you want to measure strength of strong decks, this is pretty much the only way - look at a large number of games played by a large number of people, figure out who's playing the same decks (or similar enough to group together, at least), and see. (On the other hand, determining that a deck is very weak is pretty easy; you'll tend to notice that it loses all the time.)
Perhaps most importantly, "in its format" covers up a ton of complexity. You might think that just picking a format pins everything down, but in reality there's room for a lot of variation in the metagame: what kinds of decks everyone chooses to play. Generally, a given deck will be strong against some decks but weak against others. Depending on which of those decks are being played by more people, the deck could turn out to be strong or weak.
If you're talking about casual games, the range of power is enormous, especially when people have limited budgets and are perhaps building decks to suit their preferences, not just to be competitive. A deck that's "strong" among you and those you play with might be seen completely differently by another group of people. Of course, in casual play, you're also likely to be a bit less concerned with "playable" and eking out small edges in win probability, and more concerned with just having fun.