This is three part:
[ul]
[li]What is the smallest particle (with mass) that we are aware of?[/li][li]How did we find this particle(with mass)?[/li][li]Does there have to be a smallest particle(with mass), or can it just go on forever, each particle being smaller then the last, and yet still having mass?[/li][/ul]
Is your criterion solely mass, or were you looking to define it more than that? The size of tiny particles such as quarks isn’t well-defined, so that may be difficult (though I think the mass/energy is measurable, by calculating effects on other particles.)
As for part 3, philosophically there doesn’t have to be a smaller particle (of course, all particle interactions could be invisible magic elves for all we know), and I don’t think there has to be a measurably smallest particle either, or at least we have no way of proving that there is one.
Sorry to hijack, but does anyone know who sang this song? I heard it once on the radio and it starts like this :
“Well, the smallest thing that’s known to man is a subatomic particle which is approximately 10[sup]-18[/sup] meters (C’mon, everybody now!)”
The singer then goes on about how the smallest thing that’s known to man is actually the “line between the hot and cold on my shower”, and how, when showering with his girlfriend, she had slipped and switched it from hot to cold, and a certain part of his anatomy broke all previously known records for “the smallest thing that’s known to man”.
It was a solo singer with guitar, and had kind of a folk/country sound.
[/hijack]
The smallest particles are the six quarks and six leptons. The Up quark and the down quark (together with electrons) make-up ordinary matter. The Up quark is the smallest of the quarks. The rest is usually only found in particle accelerators under extreme conditions. Quarks are called Up, Down, Strange, Charmed, Bottom, Top (getting bigger as we go).
They found the particles first based on theory. As various theories were proposed they made predicitons that certain things should exist. Scientists then ran to their particle accelerators (aka Atom Smashers) and proceeded to bang things together at ever higher energies. The higher they went the more of the things predicted by the theory were found so their faith in the theory went up. I think only one quark remains to be found but it requires stupendous amounts of energy to get to. The Superconducting Supercollider in Texas was supposed to have been able to find the Top Quark but the project was cancelled.
You cannot go on forever getting smaller and smaller particles. At the very least there is the Planck Length (1.6*10[sup]-35[/sup] which is about 10[sup]-20[/sup] times the size of a proton) that is the lower limit on size. Our quarks are a lot bigger than this but IIRC quarks are believed to be the end of the line. Unfortunately I forget the reason which isn’t much help to you but if no one else provides the answer I believe I found it in Stephen Hawking’s A Brief History of Time where he addresses this very question (good book to read in any case).
The Planck Time is the smallest amount of time that can be measured (the time it takes light to cross the Planck Length). However, I once heard the smallest conceivable measurable time in the Universe is the time it takes for the light to turn green and the cabbie behind you to honk 
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I did a search on Audiogalaxy, and came up with The Smallest Things Known to Man, by Loren Elliot. Is this what you were talking about?
To add to Whack-a-Mole’s answer, the particle with the lowest mass would be one of the neutrinos. There are three types of these - the electron, muon and tau neutrinos - and they’re three of the six leptons Whack-a-Mole mentions. Exactly which of these have to have non-zero masses on the basis of the available experimental evidence still seems a little murky, but physicists would now be gobsmacked if any of them was massless - or that the electron one wasn’t the lightest.
This page has lots of links to the experiments being done and being planned that will help determine the masses.
The argument that Whack-a-Mole is thinking about for quarks (and the leptons) being the bottom-level may be the one that has to do with binding energy. Roughly, when sub-atomic particles stick together some of their mass goes missing; this is equivalent to the energy that’s holding them together (as in E=mc[sup]2[/sup]). For things like protons and neutrons in atomic nuclei, this binding energy is of order a few percent of the mass of the particles, but for the quarks making up the protons and neutrons, their binding energies are a good chunk of their whole mass. Since the binding energy cannot be greater than the original masses of the constituent particles, it’s thus been argued that you’re unlikely to have anything smaller, interacting more strongly. Suggestive, but it doesn’t conclusively rule out lower levels of structure.
There is also evidence - from the unification of running coupling constants at high energy - that Nothing Much happens between the quark-lepton level and the Planck scale. That’d strongly constrain possible levels in between.