Ukraine used to be part of the Soviet Union (of which Russia was the main member and main power).
While part of the same 'family' of states, people from one state were free to live in others states. This made a lot of sense because, under the central planning that is typical of communist political systems, the needs of the economy and of security could best be met by people relocating.
The Soviet Union fell apart about 20 years ago and former member states became independent of each other again. Some (like Russia) stayed fairly communist in their outlook while others (like Poland) adopted a more market driven economy.
Some of the former members of the Soviet Union wish to adopt a traditional western version of a capitalist economy and, to derive the greatest benefit from this, wish to join the European Union, primarily to be part of the EU trading bloc but also because it offers them some political protection from their former, more powerful neighbour, Russia.
Ukraine's government decided to seek an alliance with the European Union. Many people living in Eastern Ukraine were of Russian decent and wanted to renew ties with Russia. Their sense of loyalty remains towards Russia and their political preferences remain more communist than capitalist. They oppose closer ties with Europe and formed a 'rebel army' to seize eastern Ukraine.
Russia sent its army and military equipment to help the Ukrainian rebels who are mostly of Russian decent and whom Russia considers its citizens.
This was an act of war, to send armies over the border into a neighbouring independent state.
The reasons Russia did this are partly political, to show that it is still a world power; partly for domestic political reasons (a weak leader can make his position stronger by creating or stressing an external threat to the nation), and partly strategic because eastern Ukraine has a vital sea port that Russia needs access to in order to maintain its military power elsewhere.
Russia calls its invasion of eastern Ukraine 'humanitarian' to protect 'it's' citizens (even though they are now Ukranians).
Ukraine calls the Russian action and act of war.
Ukraine also needs access to east coast ports for economic reasons.
The rest of the world cannot stand by and let one nation invade and bully another nation but it does not want to go to war with Russia.
For the time being, most countries in the world have applied economic sanctions against Russia in the hope this will convince it to scale back its aggression. The USA is one of the countries applying political and economic pressure on Russia.
There is no easy solution because the people of Crimea (the eastern Ukraine province) actually WANT the Russians to be there and actually WANT to part of a soviet style bloc with Russia. Most of them have not been taken by force though, of course, there are many thousands of unwilling people who are caught up in this and who are trapped in eastern Ukraine, unable to speak out or act against the occupation by Russia.