The most famous constant may be the speed of light in vacuum, which is 299 792 458 m/s, and we might not have noticed much if it instead were 250 000 000 m/s or 350 000 000 m/s, but there are other constants that are more critical.
If the strong nuclear force, which keeps the parts of an atom together, had been only 2% stronger, then all hydrogen would have been converted to helium, and no water could exist. If it had been only 5% weaker, on the other hand, no helium at all could have been formed, and no other elements either, so we would have been in a very boring universe with just a lot of hydrogen floating around.
The electrical charge of an electron is 1.602 176 thingummies. It doesn't really matter what the unit is. But if the charge of an electron had been slightly different, chemistry would have worked in a completely way, and there would have been no carbon to form amino acids, animal tissue or petrol for dad's car.
So aren't we really lucky that the physical constants happen to have the values they have?
Well, those of us who are alive today are lucky for a lot of highly unlikely events. We are lucky that Gutenberg invented printing. If he hadn't done that at that particular time, other things would have been different as well, and our ancestors in the 15th century might never have met and we would consequently not have been born.
We are lucky that the earth was struck by an asteroid 65 million years ago, because if it hadn't evolution would have taken other twists and turns than it did, and man would never have appeared in his present form. We are in fact lucky for billions of billions of events that happened in the past, and that made evolution take the path it has taken, and that made our parents and ancestors fall in love with each other. If evolution had taken a different path, it would have been very lucky for the life forms that had emerged instead, but it would definitely have been unlucky for us.
So we are in no way short of luck. In fact almost every single event in the past has been necessary to let us be born. If the Huns hadn't ravaged Europe, if Columbus had never come to America, if Henry VIII hadn't killed his wives, we would certainly not have been born. Some other people would have been born instead of me, the writer, and you the reader, and their world would perhaps have been a better one, but it would not have included the two of us.
Now, suppose that the constants of nature had been different, and there had been no hydrogen or water in the world. Who would have complained? Who would have slapped his forehead and said "Bummer! Here is not a drop of water to evolve with!" No one. No birds would be flying overhead. There would be no birds to fly. And no hands to slap any foreheads.
There may be other universes... Actually, the word "be" here is fairly nonsensical. How do you define the existence of another universe, with which you can have no contact? You cannot prove that it exists or that it doesn't. But let that be as it may. We are just playing with logical possibilities for the time being. So: There may be other universes where the physical constants are different, and where life is not possible. We are clearly not born in any of those universes. Our "luck" may simply be that there were enough other universes out there, so that at least one had the right physical constants to allow life.
The other possible answer is of course that God tweaked the constants to make us possible, but it is hardly a much more plausible answer than the one with multiple universes. You can put neither God nor other universes in an experiment, to see how they behave.
So what about the interesting things I mentioned to start with? Well, what if there are conditions for life that neither God nor different universes could have altered the probability for?
Some things are so fixed that nothing can change them. 1+1 is always 2. If you have 12 of something you can divide it in 2 equal groups of 6 or 3 equal groups of 4. That is nothing God can change. If he has 12 angels, he cannot say that 3 * 4 angels is a different number from 2 * 6 angels. And no matter how many universes you have with twisted content, if they contain 12 of something, it can be divided into either 2 groups of 6 or 3 groups of 4. There are many more mathematical facts that are as immobile as this one. The laws within Euclidian geometry are fixed. All laws of logic and the laws of mathematics are in fact all so independent of any existence, that they cannot be changed under any circumstances. They have been there all the time, and they will always be there. They are not invented. They are discovered.
Now, we can with a very large stretch of imagination imagine that something could have been incompatible between the laws of mathematics and the emergence of life. If any possible life form would have had to be based on a number between 3 and 5 that was not divisible by 2 for example. If that had been the case, God would certainly have been really miffed, when he tried to create us. "I have this great idea for an intelligent species, that would inhabit something that I would call the Earth, in something I had in mind to create and call the Universe. However, it all falls on this missing odd number between 3 and 5. Well, I guess I'll go an play some golf instead."
If anything, we do not have to be grateful for the fact that our little universe's constants happen to be right, because they could always be changed. But we should be grateful that the world of logic and mathematics is compatible with life.