Firstly, welcome to the Forum.
With all due respect to Dave, he's trying to summarise a catering-size can of worms in too few lines.
"Tiger" in relation to Triumphs was originally coined by Edward Turner when he was put in charge of the newly-formed Triumph Engineering Company in 1936. The desperate need was to sell bikes, he inherited fairly mundane 250, 350 and 500 singles, he hit on the idea of chroming a few parts like the tanks and calling 'em "Tiger 70", "Tiger 80" and "Tiger 90", the numbers alluding to the bikes' top speeds.
Turner had his iconic 500 cc "Speed Twin" ready for 1937, followed a year later by the "Tiger 100".
Following WW2, Turner essentially enlarged the 500 twins' bore 'n' stroke to create first the 650 cc "Thunderbird", followed by the "Tiger 110".
In a separate development, Triumph decided to support a factory team in the ISDT - International Six Days Trial. This was more long-distance riding on unsurfaced roads rather that what GB calls "trials" - getting motorcycles (and cars) over impossibly-large obstacles at very low speed.
When the Triumph team won the ISDT, an off-road version of - at first - the 500 twin became part of the range - called the "Trophy", a 650 version soon followed.
So, thus far, for a given engine, "Tiger" was a 'sporty' road-going version while "Trophy" was the off-road version ...
For '57, Turner developed a new 350 twin, having the engine and gearbox 'in unit' - in the same crankcase castings rather than the gearbox being separate from the engine. The following year, the engine was enlarged to 500 cc and took over from the pre-unit "Speed Twin"; another year later, the "Tiger 100" was part of the same unit-construction engine range.
Before '59, the top-of-the-range road-going 650 had been the single-carb. "Tiger 110"; '59-on, it was the twin-carb. "Bonneville" - its model code was T120.
For '63, the 650 range was redesigned with both unit-construction engines and new frames. There was still the twin-carb. T120 "Bonneville", a single-carb. road-orientated model (TR6R) and a single-carb. off-road model (TR6C) but they were both called "Trophy" ... and another a single-carb. road-orientated model, slightly differently-styled, called "Thunderbird".
In the 1960's 500 ranges, it's even harder to be precise about which name was applied to a given version - e.g. in '65 in the US, the road-orientated T100SR was called "Tiger Road Sports" and the the off-road T100SC was called "Tiger 100 Competition Trophy" ...
At the end of '66, Triumph dropped the "Thunderbird" and "Speed Twin". From '67, the twin-carb. "Daytona" became the top-of-the-range 500; "Tiger" was the single-carb. model - off-road in the US (T100C) and road-orientated in the ROTW (T100S).
In '69, the road-orientated single-carb. 650 (TR6R) became "Tiger" again ... Triumph dropped the T100S at the end of '70 and the two off-road versions - TR6C and T100C - at the end of '71. So, when the 650 twins became 750's in '73, the "Tiger" was a single-carb. road-orientated model; there wasn't another off-road "Tiger" 'til '82.
So the tigers were dirt bikes.
Uh-uh; in most years of Meriden's history, they're road bikes, and always single-carb.; "Trophy" were dirt bikes in more years, also single-carb.
Were the ISDT bikes special build?
could you get Bonnie's as off-road.
Yes. Between '64 and (iirc) '67, T120C and T120TT were twin-carb. off-road versions available in the US.