When you play tournaments you invariably play with plastic sets (not heavy), "triple weighted sets" are usually wooden sets, upper end of the market (heavy pieces). I suppose masters sometimes play invite tourneys where the standard set is a good wooden set, but not for the rest of us. I don't know about the precise equivalence of the two terms you present, sorry.
Does "Tournament Weighted" pieces = "Triple-Weighted" pieces?
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Thanks for the heads-up on that! The term "Tournament-Weighted" did sound a little too glitzy and vague to me.
Triple weighted used to refer to the three lead discs that Drueke used to place in the bases of some of their sets. Other sets used two discs. Over time the term got used by other piece makers/sellers to just mean heavy. You can even find "quadruple-weighted" sets now for that matter. In the end you'll have to get a reliable weighing or actuallly heft them in person to know if they qualify as heavy or not since we can't trust the hype.
If you are buying a set for home use (not including blitz), I suggest you buy a wooden set with 4" Kings and 2-2.25" squares. Ensure the set is weighted and felted on the bases. A 2-2.25" border around the playing surface is pleasing to the eye. If blitz with friends or travel is expected, then buy the standard plastic tournament set and roll-up board (always roll face out), either way, ensure you buy a Staunton pattern set.
p.s. An afterthought concerning wooden pieces...properly varnished pieces are the only way to go (unvarnished White pieces discolour quickly).
p.p.s. You won't pull any chicks with dirty pieces! 
Ray_Brooks wrote:
If you are buying a set for home use (not including blitz), I suggest you buy a wooden set with 4" Kings and 2-2.25" squares. Ensure the set is weighted and felted on the bases. A 2-2.25" border around the playing surface is pleasing to the eye. If blitz with friends or travel is expected, then buy the standard plastic tournament set and roll-up board (always roll face out), either way, ensure you buy a Staunton pattern set.
p.s. An afterthought concerning wooden pieces...properly varnished pieces are the only way to go (unvarnished White pieces discolour quickly).
p.p.s. You won't pull any chicks with dirty pieces!
3.75-4.00" is the typical range for tournament standard pieces. I find 2.25" and 3.75"
to be a good combination of square and piece size. For 4" kings a bigger square usually, but taste varies. Varnish, if you mean that specifically and not some general coating, is usually shiny. I see a matte wax finish a lot though, and it looks good. HOS recommends a once a year paste wax treatment for their nice sets. ( I got a couple good HOS sets but so far all chicks are unpulled.)
For blitz and other casual and potentially rough play, yeah a plastic set and vinyl
board is common. I got a vinyl folding board which is great and cheap; some
boards are too cheaply made these days though and won't lay flat,
but the folding one does and is the right size and proper colors.
For blitz a little weight is handy, and a weighted plastic set is still cheap.
Something to consider.
Was looking at some chess pieces the other day somewhere and it was stated that they were "Tournament Weighted". Someone mentioned that this is the same as "Triple-Weighted" -- true?