Why are 'finite morphisms' important in algebraic geometry? And what does a module of finite type mean? Linear transformations, Group, ring, $k$-algebra morphisms and many other types of morphisms that appear throughout mathematics are more or less obvious in the sense that we can clearly see why they are defined the way they are. Then we have continuous maps as morphisms in the category $\mathrm{Top}$ that look tricky in the way they're defined, but we don't mind them a lot because we have seen them long before we learned about morphisms and they are familiar to us. 
However, I have been struggling to understand why we need to define 'finite morphisms' in algebraic geometry. What do they help us achieve? What properties do they preserve? Why are they interesting? 
We already have a notion of morphism between algebraic varieties seen as 'spaces with functions' which seems quite natural and reasonable to me as it is. Why do we need another type of morphism between algebraic varieties?
And more importantly, what does it mean for a $B$-module $A$ to be of finite type? Is it related to the concept of a finite morphism?
 A: I think you're a bit confused.  "Finite morphisms" aren't a different type of morphism.  They're a special class of the morphisms that have nice and useful properties.  It's like how we study topological spaces, but we call some of those spaces "compact", since compactness is a handy property of some spaces that helps us prove things about them.  Similarly, some morphisms of varieties have the property of being "finite", which is helpful for proving things about them.
A "module of finite type" is just another name for a finitely generated module.  This notion is indeed very closely related to finite morphisms of varieties.  In particular, if $X$ and $Y$ are affine varieties with coordinate rings $A$ and $B$, then a morphism $f:X\to Y$ is finite iff $A$ is a finitely generated $B$-module (where the scalar multiplication of $B$ on $A$ is defined via the homomorphism $f^*:B\to A$).
A: I think you're going in the wrong direction a bit here. Every example of a type of morphisms you have in your post has special classes that we're particularly interested in - isomorphisms, for instance, are the most obvious one. Mathematicians like knowing when two things are the same up to isomorphism. Similarly, injections, surjections, increasing/decreasing maps, inner-product preserving maps, diagonalizable/non-diagonalizable maps of a vector space, etc. are all certain classes of maps that behave in nice and/or interesting ways, and people talk about them because of that behavior.
Finite maps in algebraic geometry have the property that affine-locally they look like $\operatorname{Spec} B \to \operatorname{Spec} A$ for $B$ an algebra which is finitely generated as an $A$-module. This is in some sense the best possible easy-to-define finiteness criteria. One might hope that this criteria tells us that the differences between $A,B$ are not too extreme, and looking in to finite morphisms tells us that this is the case. $A$ and $B$ have the same dimension, for instance, the fibers of the map have bounded cardinality, the preimage of any affine subset under a finite map is again affine, etc. There are many results you can prove about these things, and I'd encourage you to look at StacksProject or Hartshorne or other introductory AG texts to see further examples.
A module of finite type is just another name for a finitely generated module, which is useful as talked about above.
Additionally, to attempt to help prevent some future confusion, a morphism of finite type is sort of the next available general finiteness condition. Here, instead we require that affine-locally, our map looks like $\operatorname{Spec} B \to \operatorname{Spec} A$ for $B$ an algebra which is finitely generated as an $A$-algebra. Again, this is an eminently reasonable request, and our hope is that the differences between $A,B$ are not too bad, and it's true that we exclude pathologies here like if $B$ was infinite-dimensional over $A$, whatever that means. But not all the nice things that were true for finite morphisms are no longer true - for instance, it is not necessary that the preimage of an affine set under a morphism of finite-type is again affine.
