Is there a major difference in terms of what you can photograph with different sizes of sensor? By having a smaller sensor, can you be limited in taking pictures of certain types of subject? If so, which ones?
The lens determines what size of things you can photograph. The crop factor of a lens only impacts what focal length you need for a particular field of view, but you can still generally achieve that field of view with a different lens.
The most notable difference in terms of the type of shot you can't get with a crop sensor is if you want lots of background blur. You have a much harder time getting background blur with a crop sensor due to the shorter actual focal length you use for a given field of view. An APS-c sensor will have an easier time taking a photo for a given field of view with a wider depth of field (more in focus) and a full frame sensor will have a shallower depth of field for that same field of view and will include more background blur.
Additionally, the pixel pitch (physical size of each pixel) impacts the light sensitivity of the pixel. Assuming all else is equal, a larger surface area is more likely to be hit by a photon, so the larger the sensor, the more light it can absorb for a given resolution. This makes large sensors more sensitive in low light than their smaller contemporaries, though over time sensitivity also increases (or rather noise is reduced).
Two things to consider:
smaller pixel means less sensitivity receiving a certain amount of photons (for a given silicon technology!), needs more amplification and results in more noise per pixel compared to sensors with larger pixels. This is a noise-related disadvantage (or sensitivity-related disadvantage).
For a sensor size and for a certain silicon technology there is a maximum number of pixels. (You will have to be able to operate at ISO 100 sensitivity with a given noise/signal ratio). So a smaller sensor means potentially less pixels. E.g. probably you will not see a 100 MPixel camera for APS-C these days. This is a resolution-related disadvantage.
Now, there are two more things to consider if you are thinking between full frame (FX) versus a crop-frame (DX, APS-C) body:
smaller sensors mean smaller lenses are needed - they are cheaper and less heavy than the ones having the same focal length for full-frame bodies.
or, the other way: it is easier to achieve larger effective focal lengths with crop factor sensors than with the full-frame ones.