A lot of antibiotics are able to kill a wide range of different types of bacteria.
A lot of infections are from a predictable group of bacteria - for example bladder infections tend to be a very familiar gang of criminals, and there's selection of antibiotics for that particular gang of criminals.
Often, too, even if it isn't the absolutely perfect antbiotic for a given bunch of bacteria, it will slow them down considerably, giving the body a chance to rally.
Most antibiotics work on extremely simple, basic principles. For example, bacteria, if they don't have their 'skin', simply - well - it's pretty unattractive. They simply fall apart. Literally. Without the pressure from the membrane, they just - well - schmizzle away. So a good number of antibiotics just prevent them from making that skin, or dissolve it if it's already made. The body's immunse system can then easily engulfe and destroy.
When an antibiotic doesn't work quickly, it can be due to several different things - the infection consists of a large number of bacteria, the person is doing something that inactivates the antibiotic (for example, some medicines can't be taken with certain herbs, other meds or even - fruit juices....).
It's important to know though, that it is actually rather unusual for an antibiotic to 'immediately' work. In some diseases/antibiotics you might notice a difference in as little as 24 hours, but frankly, that's rather unusual. The antibiotic may take time to get up to the right amount within the body, then take time to get into the 'compartment' it affects (a 'compartment' is a - well - nonexistent thing, it's basically 'the area of the body where this medicine has an effect').
Resistant infections are actually still unusual, despite the press. The way to not create resistant bacteria is to always take antibiotics exactly as prescribed, and always finish them, even if you start to feel well. Otherwise, a few bacteria remain, and they may learn some way around the antibiotic.
Certainly the less often people take antibioitcs, the fewer chances there are for the bacteria to 'learn' some way around the antibiotic. So it's important to only take antibiotics when it is necessary and when one has an infection that antibiotics will kill.
Antibiotics have NO effect on viruses. And a good many respiratory infections are, in fact, not bacterial, but viral. Secondary(sneak in after a virus tires out the immune system) infections that ARE bacterial, are also, actually unusual and are only more usual in people who are immunocompromised.
There are hundreds of different kinds of bacteria. There are more of them (mass-wise) than plants and animals combined. They grow just about everywhere. There are more bacteria in your body than there are cells in your body. Only a few types actually cause disease in humans.
Most of them are our friends - going around making cheese and things like that. A good many are probiotics - and help us to digest food and absorb nutrients. In the environment, in the world, there are bacteria everywhere. All over the place. Most don't make people sick - and even the 'bad' ones, to make someone sick, they have to get past an impressive array of defenses.