What is "functional data"? If you check Wikipedia, it is defined as "data providing information about curves, surfaces or anything else varying over a continuum", which is a quite broad concept. Based on my experience, there's no strict definition of functional data in the field, but in general the data should have the following properties:
The data are ordered over a continuum (time, spatial location, etc.).
The data type is consistent.
It is reasonable to assume the data are drawn from an underlying smooth function.
The data are high-dimensional.
One famous example that is closely related to classical statistics is the time series data, since the data are collected over time. However, FDA methods typically rely on different assumptions from traditional time series models and therefore can be more flexible and interpretable in many real world applications. So next time when you get some time series data, it is worth trying some FDA methods and see their performance.
Another example is the longitudinal data collected in numerous studies, which can be viewed as sparse functional data in most cases. Of course, the sparsity and irregularity makes FDA applications to longitudinal data a bit challenging, but some research efforts have been made in this direction over the past years.
The NHANES data set introduced in another post is a good example of dense functional data with many other nonfunctional variables. The physical activity data collected through the accelerometer of wearable devices are released as minute-level activity counts (AC), resulting in 1440 observations per day per study participant. You can check out the NHANES tutorial here.