For instance, it is easy to understand how Brownian motions appear: when an object receives many small shocks that are roughly i.i.d., the resulting path will look like Brownian motions.
What type of physical processes give rise to Poisson processes?
For instance, with one dimension thought of as time: what kind of physically processes would naturally have the "memoryless property" between spikes?
The Poisson distribution has only one parameter, so it can fit mean behaviour of a process but it cannot model the variance of a process unless that variance happens to coincide with the constrained variance of the distribution. Consequently, in general, the Poisson model is a bad model, and anything that you might model as Poisson you are better off modelling with a two-parameter generalisation of the distribution (e.g., the negative binomial, quasi-Poisson, etc.). Sometimes these latter distributions can be framed mathematically as mixtures of Poisson distributions, so they reflect cases where the underlying process is Poisson, but the scale of the process varies. In any case, the advantage of the latter models is that they will tend to model the mean and variance of your process fairly well, which is really the minimum you would want from a good statistical model.
In theory, the Poisson distribution is good for modelling counts of events that occur as the limit of a binomial process when the number of trials is large and the probability of the event is small. A standard physical example of this is when you go out in a rain-storm and catch drops of rain in a test-tube --- the number of drops that land in the tube can be regarded as an outcome from a large number of rain-drops where each individual drop has a tiny probability of landing in the tube. Mathematically, if you have a sequence of event indicators $E_1,E_2,E_3,... \sim \text{IID Bern}(\theta)$ (e.g., that specific rain-drops land in the test-tube) then the count of the number of events in the first $n$ trials is $X = \sum_{i=1}^n E_i \sim \text{Bin}(n,\theta)$. If you then take the limit as $n \rightarrow \infty$ and $\theta \rightarrow 0$ such that $n \theta \rightarrow \lambda$ then you get the distribution $X \sim \text{Pois}(\lambda)$.
The Poisson model is an interesting model in probability, and it gives some basic insight that we can build on with generalisations. In reality, you almost always find that the dynamics of physical processes are more complicated than the assumptions behind this distribution, and consequently, the Poisson distribution does not generally fit well. (Specifically, its variance constraint tends not to match the true variance in the physical process you are modelling.) Even in the rain-drop example, variation in the intensity of the rain during the time you are collecting rain-drops is likely to lead to a situation where the number of drops caught in the test-tube is a mixure of Poissons rather than a Poisson random variable. It would usually be modelled reasonably well as a negative binomial distribution, though other (more complex) models can be better still.
Many statistical analysts deal with this problem by using a Poisson model, but then performing an "over-dispersion test" to see if there is evidence in the data of departure from the variance implied by the model. If the model fails this test they then switch to a generalisation such as a negative-binomial model or quasi-Poisson model. My own view is that this two-step process is silly, and you are better off just starting with a two-parameter model and avoiding this test altogether (see this related question for discussion).
TL;DR Processes with a probability density of events in time or space that is constant (and independent from what happened before) have an exponential waiting time distribution, and from this you can derive a Poisson distribution for counts
The Poisson process occurs when we count events within some interval of time (or space) and the waiting time between events (or distance between events) is independent of the past history, the time from the last event (or distance till the previous event). That is, the process has memorylessness. For the probability of an event to occur within some infinitesimal small amount of time (or space) it doesn't matter what events happened before.
Many physical processes are like this (or are approximately like this).
For instance. A Geiger counter counts ionizing particles hitting some sensor. If the origin of these ionizing particles is a large ensemble and there is no interaction between particles (or only a negligible one) then at any moment in time you have a the same probability of a particle hitting the sensor independent from how long ago the previous particle was hitting the sensor.
I myself have used mass spectrometers counting aroma molecules. There are millions of these and only some of them enter the spectrometer and the sensor. When some particle hits the sensor then this is not influencing the probability of other particles hitting the sensor. There are still millions minus one left, and these are giving practically the same probability for the next event to occur within some time frame.
Ben mentions in his answer that almost no process is truly a Poisson process. This is correct. The Poisson process is an idealized situation (in the same way almost no practical process is perfectly Gaussian). In the case of the mass spectrometer, I have three deviations:
In practice, I make sure that the experiment works well and the distribution of the experimental values is close to a Poisson process. I even used it as a test (checking whether variance and mean of the signal are equal) to control whether the machine is operating well and is not in the regime of clipping or having too low signal to noise.
Consider yourself driving a car in the rain and counting how many rain drops fall on your windscreen per second. None of these raindrops has any memory whatsoever of how many of the other raindrops reached your windscreen.
$\lambda$ will depend on your driving speed.
The Poisson distribution tells you the probability of the number of events in a specified time interval if these events are taking place independently (happening of one of them doesn't affect the probability of happening of other potential events).
I think it's better to explain via an example. Suppose you have a shop, and you want to know what is the distribution of the number of customers you get in one hour. Can you model it with a Poisson distribution? Well, does receiving a customer affect the probability of receiving the next one? Most likely yes, because if there are already some customers in your shop, the next person considering entering into your shop might say this is a busy should, I'll go to another one (on the contrary, one might argue that if there are already some costumers in your shop, it will make your shop to look more attractive). The button line is having already some customers has an effect (either positive or negative) on potential future customers. Thus, in this case, you shouldn't model it with a Poisson distribution. On the other hand, if you have a huge shop with many staff it might be reasonable to assume your shop can accommodate many many customers, and thus having some customers doesn't have a bearing on the likelihood of the next person entering your shop. If that is the assumption you want to make, you can go ahead with using Piusson as the distribution of the number of customers per hour.
The Poisson distribution is appropriate for modeling the number of time an event has happened in a specific time interval. In the transportation safety field the Poisson distribution is widely used for the analysis of crashes at individual sites. For instance, imagine having data on the monthly number of crashes at an intersection for 10 years; the distribution of the data will follow the Poisson distribution.
