Why do galaxies differ so much in size, shape and activity?

Not a complete list but:

Density of matter in that region of space where the galaxy formed.

Age of the galaxy and the stars within it.

Proximity to other galaxies and/or galactic cluster.

Broadly speaking they're shaped by gravity, so anything that affect mass or distance is going to impact.

There are two main types of galaxy: elliptical and spiral. Elliptical galaxies tend to be mostly redder, lower mass stars, smaller, and a lot older. Spiral galaxies are brighter, bluer, with a lot of gas and dust in their arms. The major difference here is time; the elliptical has had the time to become a lot more uniform. However spirals and ellipticals also have variations within those classifications - this is more a product of the way in which they have formed and developed, and the initial conditions from which they were birthed (e.g. gas and dust density, dark matter, angular momentum).

Galaxies are one of my favourite things to talk about!

Size:

Mass of course determines galaxy size (the more mass the more material can be within the galaxy). This is affected by the density of matter where the galaxy formed (determined by natural variation caused by gravitational fluctuations).

Shape:

Three general classifications for shape exist: spiral, elliptical, irregular. It's unclear what exactly gives spiral galaxies their shape but two hypotheses are particularly convincing: SSPSF and density waves (can Google this for more info). It's generally accepted that elliptical galaxies are caused by spiral galaxies interacting in some kind of way to change their shape into that of an ellipse (like the Milky Way and Andromeda in the future). Irregular galaxies usually have experienced a whole host of events that have caused them to have no set structure.

Activity:

Active galaxies are those with a centre that emits radiation; this is caused by a supermassive black hole that attracts the material around it, the material that falls towards the black hole then spins, heats up, and emits radiation at energetic wavelengths. What allows this to happen is: the size of the black hole, the rate the black hole accretes material, and the density of the central region of the galaxy.

Hope that helps!

The formation and evolution of galaxies is an active area of research with many unanswered questions. The size depends on the mass distribution of gas that forms the galaxy as well as its interactions with other galaxies. Galaxies often collide and combine which can result in a bigger galaxy (The Andromeda galaxy will collide with the Milky Way in ~4.5bn years). However, these processes are very complicated and may result in a smaller or deformed galaxy. This means the specific history of a galaxy has a large effect on how we observe it today.

Galaxy interactions also affect the shape of galaxies and the combination of spiral galaxies is one theorised origin for massive elliptical galaxies. The formation of spirals in galaxies is still not settled but they are broadly described as spiral density waves that are stabilised by their own gravity. There are several different morphological schemes for classifying the shapes of galaxies and these can be used to try and understand the evolution of galaxies.

The activity of a galaxy is most commonly referring to the behaviour of the central region of a galaxy. An active galaxy is one that has an Active Galactic Nuclei (AGN) . This is thought to be caused by material accreting onto a supermassive black hole, resulting in the emission of vast amounts of electromagnetic radiation in all wavelengths as gravitational potential energy is released. In certain cases, material outflow is concentrated in a jet that reaches relativistic speeds. AGNs can also influence the evolution of the entire galaxy by regulating star formation.

This is a broad overview and a large amount of work is still being done to tackle these questions. Hopefully you can see how the totality of variations in these factors and the histories of individual galaxies can result in the diversity of galaxies that we observe in the sky.