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 , projectiveness is defined slightly more generaly than in Hartshorne (for exampleII.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, in EGA it turns out to be local on the target scheme).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.
