This is pretty obscure, even by Soviet ethnography standards, so I'm going to do my best. I'll note that the Wikipedia article is citing Nationalist Mobilization and the Collapse of the Soviet State by Mark Beissinger, so I will start there by relating what he wrote about Tuva. Beissinger himself cites Gail Fondahl's writing on Tuva in Ian Bremmer and Ray Taras' New States, New Politics, so I will also use that for additional info.

Tuvans themselves are a Mongolic people with heavy Turkic influences. Traditionally, Tuvans practice Tibetan Buddhism, and practice pastoralism, especially in the region's Altai Mountains. Pastoralism and traditional culture were negatively impacted by collectivization and antireligious campaigns in the Soviet period. Some ethnic Russian presence had been in the area since the early 19th century (the 1860 Treaty of Peking with the Qing Empire allowed Russians to pass through but not settle in the region), and the region had since that time experienced an influx of Russian miners and traders similar to other parts of Siberia. Local Qing administration could do little to stop the influx, and local Tuvans often resorted to violence, with the Russians in turn appealing to the Russian government for military protection from "native attacks and revolts". In 1885 a Russian okrug was established to deal with trade in Tuva, while a new rule in the 1870s allowed settlements of Russians as long as they numbered 200 or less (Tuvans provided these miners with foodstuffs).

After the 1911 Chinese Revolution, a number of local Tuvans appealed to Russia for additional protection, and a protectorate was established in 1914, leading to increased Russian immigration (from 2,100 in 1910 to 8,200 in 1916). The Russians increasingly saw the area as part of Russia, while Tuvans were much more divided on the preferred fate of their region - a part of China? Mongolia? Or independent? The area more or less descended into chaos in the Russian Civil War, and was fought over by Reds and Whites, while also being claimed by China and Mongolia.

In 1921 a Tuvan Soviet (basically, a local council) voted for independence of Tuva (the Tuva People's Republic, aka Tannu Tuva), under Soviet supervision, and Donduk Kuular, a Buddhist monk, was elected Prime Minister. Russians in the republic were not given citizenship, but were effectively Soviet citizens with extraterritoriality. Donduk's government declared Lamaist Buddhism to be the state religion, required religious training for young Tuvans, and supported reunification with Mongolia and pan-Mongolism. This last policy was decidedly not Moscow's favorite, and they in turn encouraged a separate Tuvan national identity, playing up Turkic roots over Mongol, and introducing a "neo-Turkic latinized alphabet" for the new Tuvan literary language. In 1929 soft power gave way to hard power, and Donduk was purged and replaced with Solchak Toka, a Moscow-trained Communist, who remained in power until his death in 1973. Russian settlers were made full citizens of the republic, Soviet specialists assumed numerous governmental positions, and collectivization was undertaken. In 1944, the Republic's Little Khural (which was their equivalent of the USSR Central Committee) petitioned for admission to the USSR, and Tuva was incorporated as an autonomous oblast in the RSFSR, before being elevated to Autonomous SSR status in the RSFSR in 1961. The rationale for the annexation was both strategic (Tuva was a "fortress guarding the approaches to the Kuzbass" industrial region), and as a means of discouraging nationalism (separatism among other peoples like the Oirats would be discouraged without an independent Tuva to look to).

After absorption into the USSR, Russian immigration increased, and by 1959, some 40% of the population was ethnic Russian. Mining was a regional source of wealth, but largely did not benefit the Tuvan population, which experienced from then and even after the Soviet period extremely high unemployment and some of the lowest living standards in Russia.

Tuva then was unique among Autonomous Soviet Socialist Republics in the RSFSR in that it was (in theory at least) an independent country from 1921 to 1944, albeit a Soviet client state, and its independence was only recognized by the USSR and Soviet-dominated Mongolia. However, when mass movements in the Baltic SSRs started organizing for the restoration of their interwar independence, apparently some Tuvans were inspired to organize for a similar goal.

This isn't totally weird. Some 64% of the Tuvan ASSR were ethnic Tuvans, with about a third of the remainer ethnic Russians in 1989. In comparison, Estonia at the time was some 62% Estonian and 30% ethnic Russian, and Latvia was only 52% Latvian and 34% ethnic Russian (the Kazakh SSR was 40% Kazakh and 38% ethnic Russian, if you're interested in the lowest titular nationality / highest Russian percentage in an SSR at the time). Of course, Tuva had a much smaller population than even Estonia, with the number of ethnic Tuvans being some 207,000 in a rural, remote and very mountainous republic. The capital city, Kyzyl, is by far the largest city in the republic, with a third of the population living there, and that means it's still a town of about 100,000.

In late 1989 a Tuvan Popular Front (Khostug Tyva or Free Tuva) was organized in direct imitation of the Baltic movements), and it gained 9 out of 130 seats in the local republican legislature in the March 1990 election. The Popular Front initially advocated for upgrading Tuva's status to a full Soviet Socialist Republic, but later turned to calling for a referendum on complete independence. More radical members wanted the Russian population out, and wanted to organize a militant national guard. In any case, the movement largely was urban based, and became moribund when its leader, Kadyr-ool Bicheldei, was elected to the RSFSR Congress of People's Deputies in March 1990, and moved to Moscow to take up his office there.

The more radical elements in the Tuvan nationalist movement turned towards more violent means, although this was basically an intensification of occasional Tuvan-Russian violence that had occurred since the 1970s. Again, it's worth remembering that Tuva was basically a relatively recently-annexed border region that was incredibly mountainous, and thus prone to lawlessness. The escalation of violence seems to have started in December 1989 when Tuvans on horseback began to attack Russian settlements and the anti-Russian murders began to take off, causing some three thousand ethnic Russians to emigrate from the republic in the first six months of 1990 (the ethnic Russian population has continued to fall, and in absolute numbers is about half of what it was in 1989).

This is where the escalation of violence mentioned in the wikipedia article comes in: there were a number of riots, attacks on trucks from other parts of Russia, and rural murders of ethnic Russians in the period with at least 88 killed "by July 1990" (Beissinger does not specify a start date) and an additional 80 killed in August, at which point several thousand OMON troops were dispatched to the republic, and order restored, if under tense politics.

In September 1991, the entire republican government was forced to resign en masse for publicly supporting the August 1991 coup plotters in Moscow, and new legislative and presidential elections were held. Kadyr-ool Bicheldei was elected Speaker of the Legislature, and a former local Communist Party Secretary and Tuvan Prime Minister Sherig-ool Oorzhak was subsequently elected President, assuming office in 1992 (he held this post until 2007).

In any case: Beissinger states that radical members of the former Popular Front tried to reorganize and press for an independence referendum in 1992, but this movement was banned from participating in local elections in 1993, and pretty much disappeared after this time. The Republican legislature (the Supreme Khural) in any case voted overwhelmingly in 1992 against the holding of any such referendum - 92% voting against. The Deputies at the time noted that the over 90% of the Republic's budget came as subsidies from Moscow, and so independence was simply unfeasible. In any case, by this time, Tuva was a signatory to Russia's Federation Treaty, and recognized as a constituent republic with its own constitution (Russia has 22 such republics). The Tuvan constitution as ratified in 1993 recognized the republic's right to secede, but even by 1994 support for this was a minority view - only 27% of Tuvan respondents supported any republic having this right, and even less - 13% - supported Tuva actually exercising this right. Beissinger writes off the secessionist movement as too narrowly urban based, and ultimately outmanuevered by more traditional nomenklatura elites in the republic.