They have some important (but not necessarily unique) properties:
The following website has an excellent summary of these key properties: http://www.chemguide.co.uk/inorganic/transition/features.html
Variable Oxidation States/Numbers
In all metals, the s-orbital electrons are involved in bonding. This is usually an ionic bond (when it is with another element). As d-orbital electrons can also be involved in bonding, there are many permutations for the number of electrons being involved in bonding, so often more than one possible oxidation state/number.
Zinc is an exception, as its ion has a full 3d-orbital. This makes these electrons unavailable for bonding, similar to the p-orbital electrons of Noble Gases being unavailable for bonding.
This is summarised well in this website: http://www.s-cool.co.uk/a-level/chemistry/transition-metals/revise-it/variable-oxidation-states
Coloured Compounds
Substances are coloured when they absorb part of the electromagnetic spectrum (light) and reflect the remainder. Transition metals' ions are usually coloured because they have incomplete d-orbitals. When bound to other ions/molecules, the d-orbital splits into two groups (with slightly different energy levels). The energy levels of these is so close that white light has enough energy to excited electrons from one energy level to the other. The remaining wavelengths/frequencies are the colour we see.
Zinc is an exception, as its ion has a full 3d-orbital. This makes it impossible for the electrons to move into an excited state due to electromagnetic radiation (light) as they would need to "jump" into a completely different orbital or energy level.
This is summarised well in this website: http://www.s-cool.co.uk/a-level/chemistry/transition-metals/revise-it/coloured-compounds
Catalysts
Simplistically, the density of positive charge in transition metal ions, and their ability to make dative bonds (due to incomplete d-orbitals), make these useful catalysts.
Alternatively, the variable oxidation states of transition metals allow them to be involved in the reaction (alternate pathway provided) then be recovered in their original oxidation state. While being involved, the transition metal is not "used up", so this is still considered to be a catalyst.
There are some excellent examples in these websites: http://www.sky-web.pwp.blueyonder.co.uk/Science/transitionmetalcatalysts.htm and http://www.4college.co.uk/a/ss/catalyst.php
Complex Ions
Explaining the details of this is actually very complicated. However, the reason transition metals can form complex ions is very similar to their ability to be catalysts; a dative bond can be formed with molecules/ions (ligands) due to the charge density of the cation. d-orbitals are involved in this, but the d-orbitals do not need to be incomplete.
This is well explained in this website: http://www.s-cool.co.uk/a-level/chemistry/transition-metals/revise-it/complex-ions
A more in-depth explanation can be found on these pages:
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