I am not too certain what these two properties mean geometrically. It sounds very vaguely to me that finite type corresponds to some sort of "finite dimensionality", while finite corresponds to "ramified cover". Is there any way to make this precise? Or can anyone elaborate on the geometric meaning of it?



Finite type/finite morphism
I definitely agree with Peter's general intuitive description.


In response to some of the subsequent comments, here are some implications to keep in mind: 


**Finite** ==> **finite fibres** (1971 EGA I 6.11.1) and **projective** (EGA II 6.1.11), hence **proper** (EGA II 5.5.3), but *not conversely*, contrary to popular belief ;)


**Proper** + **locally finite presentation** + **finite fibres** ==> **finite** (EGA IV (part 3) 8.11.1)


When reading about these, you'll need to know that "quasi-finite" means "finite type with finite fibres." Also be warned that in EGA (II.5.5.2) projective means $X$ is a closed subscheme of a "finite type projective bundle" $\mathbb{P}\_Y(\mathcal{E})$, which gives a nice description via relative Proj, whereas "Hartshorne-projective" more restrictively means that $X$ is closed subscheme of "projective n-space" $\mathbb{P}^n\_Y$.


When the target (or "base" scheme) is locally Noetherian, like pretty much anything that comes up in "geometry", a proper morphism is automatically of locally finite presentation, so in that case we do have 


**finite** <==> **proper** + **finite fibres**


Regarding "locally finite type", its does *not* imply finite dimensionality of the fibres; rather, it's about finite dimensionality of small neighborhoods of the source of the map. For example, you can cover a scheme by some super-duper-uncountably-infinite disjoint union of copies of itself that is LFT but not FT, since it has gigantic fibres.


