Can COVID-19 affect a person a second time?
Short answer: Although there are some reports on this, it is pretty unlikely. It is more likely that patients where released too early from hospital, developed further symptoms later on and got worse, was re-hospitalized, tested again for SARS-CoV2, which was positive and counted as re-infected. 

Another possibility is that a false negative test happened, when people got re-tested later the test was positive again and the patient was counted as re-infected. Given the rising number of cases worldwide this should become evident pretty soon, if this is a real problem and not an artefact. For the moment, I don't think there is enough evidence for the re-infection of the same patient in such a short time period. If this holds, future will tell.

Long answer: There is now an experimental paper, which infected rhesus macaques with SARS-CoV2, tested if the infection was similar to humans happening in the upper airways (which it was), and re-infected the animals after 28 days with a really high dose of virus (which is much higher than what comes around in our environment), with none of the macaques getting re-infected or showing pathological signs of the infection. See reference 1 for details.
This article has its limitations with small sample size and the general questions if the chosen animal model is really comparable with human. 

Additionally it looks very much like that patients who are recovering don't shed infectious virus anymore. The PCR test however does not discriminate between infectious virus and plain virus RNA, so it is most like much longer positive than people are infectious. They also show the rapid appearance of specific antibodies, which would at least temporarily protect against a re-infection. See reference 2 for a nice summary and reference 3 for the original article.

References


Reinfection could notoccur in SARS-CoV-2 infected rhesus macaques
People ‘shed’ high levels of coronavirus, study finds, but most are
likely not infectious after recovery begins
Clinical presentation and virological assessment of hospitalized
cases of coronavirus disease 2019 in a travel-associated
transmission cluster