A "stable" system is one whose state will not change spontaneously in an interesting period of time. An "unstable" system is one where the state change is too likely to be ignored in your calculations. Sometimes people use "metastable" to describe systems whose incipient state change is much slower than another interesting process.
These are plain English words. If we say that a building is stable, we mean that it is likely to remain a building for the foreseeable future, rather than spontaneously collapsing into a pile of rubble. If you have ever visited a construction site, you have been told "don't walk on that section there yet, it is still unstable." But "stable" has an implicit timescale. A house might be stable for decades or centuries without active maintenance, but rare processes like termites or tropical storms or urban redevelopment or warfare mean that the house you are currently in, right now, is unlikely to still look like a house in a hundred years. Last week the Gazientep Castle in Turkey was destroyed in an earthquake, after standing for 2000 years.
In a quantum-mechanical system, there is one "ground state" whose energy is lower than any other "excited" state. If the system finds itself in an excited state, it can release energy and increase entropy by transitioning to the ground state. But in some cases, there is some symmetry which forbids the transition. If the transition time is just very long, we call the excited state "metastable"; if the symmetry is exact and we have reason to believe the transition will never happen, we call the system stable.
For example, tantalum-180 is beta-unstable with a half-life of a few hours. There is a nuclear excited state, Ta-180m, which could in principle decay to the ground state by emitting a photon — but that photon would have very low energy (which generally makes transitions slower) and would also have to carry away an angular momentum of $\hbar\Delta J=8\hbar$. There is a small term in the transition probability which goes to the $\Delta J$-th power, and $(\text{small})^8$ is very small. Based on searches for decays which came up empty, the lifetime is at least $10^{15}$ years. If you dig tantalum out of the ground, some of it is Ta-180m. If you had made some Ta-180m at the beginning of the universe, at least 99.999% of it would still remain un-decayed. "Unstable" or "metastable" just don't feel like the right descriptions of this material.
Two other symmetries which limit decays are the conservation of "baryon number" and "lepton number." A baryon is a proton or a neutron, or an excitation which decays eventually into a proton or a neutron; a lepton is a particle, like the electron or its neutrino, which is blind to the strong force. We know of no process that changes baryon number or lepton number. (Antibaryons and antileptons with negative numbers, so pair production doesn't count.) We suspect that such a process must exist, because the universe contains more baryons than antibaryons. We also suspect that such a $B$-violating process might still take place down in the bowels of the proton, allowing processes like $\rm p^+\to \pi^0 e^+$. The Super Kamioka experiment was a skyscraper-sized vessel of water buried in a mine in Japan. After years of searching unsuccessfully for a proton decay event, the claim now is that the proton's lifetime is no shorter than $10^{33}$ years. If every minute since the beginning of the universe were stretched to the current age of the universe, that would still be shorter than the mean proton lifetime. Unfathomable time. The question of whether proton is absolutely stable is important for understanding how our universe came to be filled with enough protons that they came together and formed introspective humans. But at the same time ... the proton is stable. The proton is the ground state of the QCD vacuum, when the baryon number is nonzero. Protons aren't going anywhere.
The question of whether any particles are absolutely stable is a profound and tricky one, because more and more rare processes come into play as you look further into the future. There is currently a comment under your question which claims that all nuclei will eventually evolve into nickel-56. That's wrong. Nickel-56 (or whichever isotope, writing on a phone stinks) is the most tightly bound nucleus, but, as for the tantalum ground state, there isn't any pathway to get there. The processes which form nickel-56 are more likely to form iron, as evidenced by the fact that Earth's core is more iron than nickel. But the stellar processes which form iron and nickel also require a large amount of non-iron and non-nickel to function. Too much iron in a star, and the iron transitions via supernova to degenerate neutron matter, or to a black hole.
There is good reason to suspect that, on very long timescales, the matter sector of our universe will evolve towards being entirely black holes. It is also believed that, once its environment is cold enough, a black hole becomes unstable against photon emission, and begins to radiate away more thermal energy than it absorbs — which makes the black hole hotter, which makes it radiate faster, until it evaporates completely. (Its final instant also includes some massive-particle radiation; very exciting.) So a universe containing only black holes would evolve towards a universe containing only photons. We believe that the photon is a stable particle, but it is not clear whether a universe which contains only photons is stable; the accelerating expansion of our universe suggests that such a spacetime may be dominated by dark energy, whatever that is, and may undergo inflationary expansion.
Shortly after the discovery of the accelerating expansion of the universe (so, probably 1999 or 2000), I remember reading an article in Scientific American about the possibility of a universe which would undergo not a "big crunch" or a "big rip" but instead remain a flat spacetime basically forever. One point that the authors made was that, if our universe is currently about $10^{17}$ seconds old, then in the very distant future it would make sense to consider one $10^{17}$-th of the age of the universe as a "brief" interval. The article had a logarithmic timeline with events like "all matter is black holes" and "all black holes have evaporated." Somewhere around $10^{100}$ or $10^{150}$ years into this timeline was the phrase "quantum tunneling liquifies matter." Stability in this eternal sense is really an open question.
If the lack of a sharp dividing line between "stable" versus "unstable" bothers you, as it bothers the Wikipedia editors who wrote your quoted sentence about tungsten-180, I suggest you read an essay by Dawkins about "the tyranny of the discontinuous mind." It is the chapter in his The Ancestors' Tale where he discusses "ring species."
