TL;DR Eurovision is like European American Idol except 100% better and Australia is also there. Melodifestivalen is Sweden's national final to pick their song for Eurovision.
From Wikipedia
The Eurovision Song Contest (French: Concours Eurovision de la chanson), often simply called Eurovision, is an international song competition held primarily among the member countries of the European Broadcasting Union. Each participating country submits an original song to be performed on live television and radio, then casts votes for the other countries' songs to determine the winner. At least 50 countries are eligible to compete as of 2018, and since 2015, Australia has been allowed as a guest entrant.
Eurovision started in 1956, and has been aired every year since!
Any country that has a TV broadcaster with an active European Broadcasting Union membership. So this membership includes the European countries you'd expect, like France, Germany, Italy, etc. but also includes the deeper eastern European countries likes Montenegro, Slovakia and then countries like Israel, Morocco, etc.
Australia gets to go to Eurovision due to how they've been watching Eurovision for the last ~30 years, as a result of the large European expat population there.
The EBU doesn't dictate one method of picking their song. However one of the important hard and fast rules: the song can not have been released before 1 September of the year prior to the contest. So for the 2019 contest, songs released after 1 September 2018 are eligible.
There are two primary selection methods: internal broadcaster selection (so the country's participating broadcaster picks a song and an artist) or, the more popular method: hold a national final to pick the song via televoting, a national/international jury, or some combination of both.
Sweden's national final, Melodifestivalen, is arguably the most famous national final.
There are two semi finals, and then a Grand Final. Only 25 countries get to participate in the Grand Final.
France, Germany, Italy, Spain, and the United Kingdom are a part of The Big 5, the biggest financial contributors to the Contest. As a result, they get a direct ticket to the Grand Final.
The remaining 20 spots get awarded to the top 10 countries of both semi finals. Countries are assigned to a semi final based on the Semi Final Allocation Draw:
The thirty-six semi-finalists had been allocated into six pots, based on historical voting patterns as calculated by the contest's official televoting partner Digame. Drawing from different pots helps to reduce the chance of so-called neighbourly voting and increases suspense in the semi-finals. The draw also determined the semi-final that each of the six automatic finalist countries (host country Israel and Big Five countries France, Germany, Italy, Spain, and the United Kingdom) will broadcast and vote in.
Note: the actual running order of the songs for the semi finals and the Grand Final is now determined and proposed by the production team of that year's Contest, and is approved by the EBU.
From Wikipedia:
Melodifestivalen (Swedish pronunciation: [mɛlʊ²diːfɛstɪˌvɑːlɛn]; literally "the Melody Festival") is an annual song competition organised by Swedish public broadcasters Sveriges Television (SVT) and Sveriges Radio (SR). It determines the country's representative for the Eurovision Song Contest, and has been staged almost every year since 1958. Since 2000, the competition has been the most popular television programme in Sweden; it is also broadcast on radio and the Internet. In 2012, the semi-finals averaged 3.3 million viewers, and over an estimated four million people in Sweden watched the final, almost half of the Swedish population.
SVT has an open call for song submissions and from those songs picks 32 of them, to be assigned to four semi finals, eight songs in each. Then there is a second chance round, and then the final.
From each semi final, the top two songs go direkt till final, claiming eight of twelve spots. 3rd and 4th place will go to the Second Chance round, and will pair off in four televote duels. The winners of each claim the remaining four spots.
Then, the winner is decided by 60% Swedish televote, and 40% international jury vote, where Sweden invites some people from other competiting Eurovision countries to help quality check the song. This jury vote alone is why Melodifestivalen is so famous: it serves as a Eurovision simulator for the final in May.