TI was one of the first semiconductor companies, and is hugely well established, they acquired Burr-Brown (specialist analog house) some years back and more recently Luminary Micro - the first ARM-Cortex M3 implementor. TI is possibly the largest Digital Signal Processing chip maker and is also the originator of the DLP digital projector technology. Their ARM Cortex-A8 and A9 chips are used in cell phones and console products. Back in the 60's the integrated circuit was invented basically simultaneously by TI and Fairchild.
National was spawned by Sperry with a group of Sperry staff creating National in Danbury CT. In 1965 they acquired Molectro, and gained as employees Dave Talbot and Bob Widlar who were ex-Fairchild and possibly the leading IC designers of the era. In 1967 Chrlie Sporck was hired away form Fairchild and once again the bedevilled raltionship between Widlar and Sprock turned to fire. Then in 1968 National moved to Santa Clara and was well known for poaching staff from Fairchild. with designers like Bob Widlar they soon took the analog market by storm, they were a massive power house in discrete transistors (not power), CMOS logic and specialty chips including a number of early micros. In the end they swallowed up Fairchild (1987) and eventually spat it out again in 1997.
TI ranks 4th in the industry with 4.3% market share and about $30bn in revenue. National now ranks down there, with $1.42bn of revenues last year. The big thing is the massive consolidation of commodity semiconductors houses. In recent years TI and National have slugged it out in the commodity chip market, now we will have to wait and see what synergies work out and which product lines disappear. Two of the great names now rolled into one maybe!