The vaccine doesn't set up a force field round you that stops the Covid virus reaching you. It sets up your immune system so that, when you get the virus in your nose and/or throat, your defences swing into action much quicker than if you weren't vaccinated to destroy the virus.
According to how much virus you have on board, how many antibodies you have circulating, how quickly your system can produce all your defence mechanisms and how effective they are, you may conquer the virus almost immediately, in which case you would only test positive for a very short time, if at all, which may not be a time when you test, or you may have virus present for a while and test positive. Or, again, you may destroy the active virus, but test positive on PCR for quite a while because it picks up inactive genetic material.
If your defences overcome the virus promptly you may hardly feel unwell, or not notice any symptoms. If they take a while you may become unwell to some degree while they do their work. If you are unlucky enough that they struggle, you may develop severe disease, but even then, possibly with medical support, most people eventually win out. But, as with any infection, some people, unfortunately, just can't beat the virus and it claims their life, frequently in conjunction with another medical problem.