The Flugelhorn (or "fluegelhorn") was originally used on the battlefield to summon flanks of soldiers – a far (and perhaps, less-dramatic) cry from it's current use as the horn of choice when commercial trumpet players want to play a soft ballad or smooth jazz tune! In my humble opinion, I think the horn itself is better served in its latter role – it's hard to imagine you'd even hear a soft flugelhorn over a modern-day fighter jet!
Like most players under 50, my first introduction to the horn was with the music of Chuck Mangione, particularly his late 1970's mega-hit, "Feels So Good". But several notable players made good use of it in earlier decades. Shorty Rogers, Art Farmer, Clark Terry, Miles Davis, Chet Baker, Freddie Hubbard and others put their trumpets down on occasion to take advantage of the flugelhorn's softer, darker and mellower sound (its earliest known jazz use came from Joe Bishop, a member of Woody Herman's band in 1936).
Today, a professional trumpet player on the jazz or commercial scene is pretty much expected to double on the instrument. In a horn section, having your trumpet players switch to flugelhorn can be used to great effect, and it is often the instrument of choice for soloists looking to explore the softer side of their playing.
Playing the flugelhorn is essentially the same as playing trumpet – same basic technique, same fingerings, same key, albeit with a similar but less expansive range. It may also require a little more air to get the full sound, given its larger conical bore. There are also various versions of the horn, including rotary valve models and models which include a fourth valve (bringing the horn down a fourth or replacing the first-and-third valve combination).
However, how you approach playing the instrument is crucial. Many players tend to play the horn with the exact same air pressure, energy and technique as the trumpet, often over blowing and trying to play the horn like a trumpet. I believe this approach loses sight of why you'd even switch between the two instruments. The horn, I believe, is at its best when approached with a more relaxed and even quiet mindset. For me, if you're playing the horn loudly or in the same range as a lead trumpet, you might as well be on trumpet! Playing the horn differently than the trumpet offers you a far wider palette to choose from when expressing yourself musically.
The best thing you can do to begin exploring the horn is to listen to the great players, including the ones listed above and many others. You will often hear a marked difference in their sound and approach. And the instrument is not limited to the jazz world - Burt Bacharach made great use of its 'poppy' section sound in many of his hits from the 1960's. There is also a wealth of great flugelhorn music in more traditional styles, from symphonies by Mahler and Stravinsky to Brass Bands to Drum and Bugle Corps.
Picking the right horn, well, that's a whole other issue. Just like finding your perfect trumpet, it's going to take trying a lot of different models to land on just the right one for you – that special horn that feels more like an extension of your body than a piece of plumbing. I will offer some of the brands (if not the models) that are popular among players, but please don't limit yourself to this list. Try as many as you can!
Have fun exploring this wonderful instrument, and remember, keep it smooth and pretty – you don't want to accidentally summon a platoon!