In particle physics (as with most sciences) we are rarely ever concerned with analysing single events. What we look at are distributions of the same measurement(s) made many times. From performing fits to these distributions, we can infer the number of 'signal' and 'background' events in our sample.
Signal is the process we're interested in.
Background is all of the events which end up in the sample but aren't a result of the signal process. Often the largest source of background is combinatorial: i.e. random combinations of particles which happen to end up looking like the signal. Other sources include things like misidentified or misreconstructed processes.
If we're looking at the invariant mass distribution of a particular final state, the signal will often be a peak-like shape. Combinatorial backgrounds tend to follow smooth broad shapes (like exponential or simple polynomial curves). However, a statistical fluctuation in the background may result in a peak-like shape. It is important not to confuse this with signal. Statistical significance is a measure of how unlikely it is, assuming the null hypothesis, that a fluctuation will result in a peak at least as big as the one you observe. In particle physics, when you think you've found something, it's important to quote the significance of the measurement. The standard for discovery is $5\sigma$, which corresponds to a probability of 1 in 1.7 million.
It is important to note that we can never really tell exactly which particular events are signal and which are background. However, you can apply selection criteria to remove the background-like events and hopefully retain most of the signal. These can range from simple cuts on variables to training some complicated machine-learning algorithm to distinguish signal from background.
Let's take the example of searching for the Higgs boson decaying to a pair of photons ($H\to\gamma\gamma$). You start with a sample of recorded events containing two reconstructed photons and apply some selection criteria to filter out the background events which look least likely to be Higgs decays. If you plot the distribution of the invariant mass of the photon pairs, you end up with something that looks like the figure below:
We know a priori that the combinatorial background follows a nice smooth shape, so this is modelled as some sort of polynomial function. The signal will appear as a bump in the distribution, centred at the mass of the Higgs (125 GeV). This looks to be modelled as a bifurcated Gaussian of Crystal Ball function or something similar.
The fit finds a significant peak above the background at the mass of the Higgs. From this, you can say (with some quantifiable uncertainty) how many $H\to\gamma\gamma$ decays and how many background $\gamma\gamma$ pairs are in the data, but not which events are actual Higgs bosons or background events.