Indium is a metal in Group 13 (or III if you're old school) of the periodic table. Because it's near the metalloids it is somewhat like a classic metal and a bit like an alkali metal. For example, it is a shiny solid at room temperature but is also easily cut and will melt at a few hundred degrees C (see above writeups for physical details). For comparison, iron (a classical transition metal) melts at 1538 C.
Indium was used during WWII for coating high performance ball bearings and later for lead-free solders. It has also been used for corrosion resistant mirrors (normal silver backed mirrors tarnish faster) and has myriad uses in the elctronics industry as several indium compounds are useful semi-conductors.
For an organic chemist, indium's most attractive properties are the ability to mediate carbon-carbon (C-C) bonds in water. Making C-C bonds is essentially all of organic synthesis (apart from adding functional groups). The classic way of making C-C bonds is using a Grignard (organomagnesiumhalide) or organolithium reagent. These require anhydrous conditions which are, frankly, a pain in the arse. It involves drying all the glassware (often in an oven for several hours), freshly distilled solvent and an inert atmosphere (flooding the glassware with nitrogen or argon). In contrast, indium mediated reactions can often proceed in, and in some cases require, a water-organic solvent (usually THF or DMF) mixture. For example, formation of cyclopropanes, very strained molecules, can be achieved in a water:DMF mixture in relatively high yields depending on other groups attached. This is especially useful in industrial situations where purging a 20 gallon reaction tank of any remaining water is not really an option (due to the inordinate amount of time and money it would cost)
Also, aqueous indium can often be reused for the next batch (saving costs of extracting and re-purifying it) eg. an indium mediated IntraMolecular Diels-Alder (IMDA) reaction was run in a 6:1 mixture of H2O:iso-propanol in 83% yield. After washing the reaction mixture with diethyl ether and extracting the organic phase, the aqueous phase (containing the indium catalyst) was reused and gave a yield of 78%. The same procedure was repeated and gave a third yield of 79%.
All of these factors combine to make indium and its compounds excellent additions to the industrial process chemist's bag of tricks. Any aqueous reaction is always good but especially when considering scaling the process up (anhydrous techniques work well in gram scale syntheses, not so well on the kilogram/ton scale). Coupled with not needing to treat the catalyst with anything but diethyl ether (very cheap) to regenerate it, indium is becoming more and more attractive as its properties and synthetic values are explored and more widely recognised.
(I will add references soon, I'm in the middle of a project for my degree and taking a break during an all nighter)