The following uses for titanium are gathered from a number of sources as well as from anecdotal comments. I'd be delighted to receive corrections as well as additional referenced uses (please use the feedback mechanism to add uses).
Titanium metal is used for alloys with with aluminium, molybdenum, manganese, iron, and other metals. These alloys of titanium are used principally in the aerospace industry, for both airframes and engines, where lightweight strength and ability to withstand extremes of temperature are important. Titanium is as strong as steel, but much lighter. It is twice as strong as aluminium. It is nearly as resistant to corrosion as platinum.
Titanium is a component of joint replacement parts, including hip ball and sockets. These may last for 20 years or so. Titanium is used in dental implants because it is able to "osseointegrate" (an unusual ability by which titanium fuses with bone tissue, perhaps through the titanium oxide layer on tianium metal). This results in implants that, while not cheap, can last 30 years.
It has excellent resistance to sea water and is used for propeller shafts, rigging, and other parts of ships exposed to salt water. A titanium anode coated with platinum provides cathodic protection from corrosion by salt water. Titanium paint is an excellent reflector of infrared radiation, and is extensively used in solar observatories where heat causes poor viewing conditions.
Pure titanium dioxide is relatively clear and has an extremely high index of refraction with an optical dispersion higher than diamond. It is produced artificially for use as a gemstone, but it is relatively soft. Star sapphires and rubies exhibit their asterism as a result of the presence of TiO2. The dioxide is used extensively for paint as it is permanent and has good covering power. Titanium oxide pigment accounts for the largest use of the element.
Titanium is extremely elastic (springy), lightweight, corrosion resistant and non-magnetic and this results in new uses such as for money clips used to hold bank notes and even credit cards (as titanium is non-magnetic it has no effect upon the card's magnetic strip).
Titanium is normally coated by a very very thin oxide layer. This layer can be thickened through an anodization process to give a product whose perceived colour colour may be tuned by controlling the thickness of the oxide. Titanium is also used for jewellery (titanium rings and earrings), despite difficulties working it, because it is regarded as hypoallergenic and does not discolour skin.