It started with Ukraine's financial problems. Ukraine was trying to work out a deal with the IMF but Russia offered them a $15 billion bailout. The bailout included subsidies for oil. Ukraine does not have its own independent source for oil and actually depends on the Russia to provide it. You may be familiar with Russia turning off Ukraine's supply of oil many times in the past. The majority of people in Ukraine wanted to work with the European Union however Russia's influence on Ukraine (because of the bail out, and the oil subsidies, as well as threats to cut off all access to oil) made the Ukrainian government side with Russia as opposed to working out the trade deal with the European Union.
The people of Ukraine were extremely upset and protested. Eventually protests that were peaceful turned violent. Some of the protests where co-opted by Neo Nazi organizations, and other extremely right wing (and violent) individuals.
The government then made a series of anti-protest laws that were simply ridiculous. For example, simply protesting in front of a building and making it harder for people to enter that building can get you 6 years in prison. If you gather with a group and simply talk negatively about certain members of the government you can now get as much as 2 years in prison. The laws had the opposite effect and made the protestors even more violent.
Within a matter of days the laws were repealed and eventually the protesters successfully ousted the prime minister (who now has been seen in Moscow). The government started negotiating with the protestors. Progress and financial independence from Russia seemed inevitable. This made Putin very angry because this meant that Ukraine would switch their allegiance from Russia to the European Union and the IMF.
Putin wants to create a post communist Eurasian union which Kazakhstan and Belarus have already agreed to join. Many believe that this union is simply a disguise for combining all the post-communist countries into one huge organization resembling the USSR once again. This is the crux of the protesters argument.
Putin believes that even though he has gained support for this union in other post communist countries, the protests in Ukraine might remove some of the successes he has gained. Furthermore, this could potentially stop other post communist countries from joining the union, thus he is putting military pressure to ensure that the protests do not leak to other post communist Eastern European nations.
The WWIII aspect plays into this because Ukraine is requesting NATO support, which the US is part of, but this is not just limited to United States, Ukraine and Russia. NATO consists of 28 sovereign countries that have agreed to support each other militarily in case they are invaded. Many of those countries have other alliances which would increase the number of nations involved in any potential military intervention. The US has warned Russia as has have many other countries that their actions "have consequences".
The question now is what will Russia do? If they don't leave will NATO take military action against Russia? If so, will China support Russia? Pretty soon this could escalate to into war with 35+ countries engaging in military action.
Personally, I don't think we'll get there - but it is a real risk, and one that needs serious thought on how it can be avoided without Putin having to go back with his tail between his legs. If he can't save face this can start another Cold War.