Welcome
First of all we'd like to give you a warm welcome and hug for joining our community. Let us explain a few quick things before you start your journey.
The Tutorial
If you are completely new to osu!, it is recommended that you play the tutorial first (Peter Lambert - osu! tutorial). After Installation osu!, there will be a massive osu! button in the middle. Just click on it twice and choose "Solo" mode. You can easily search for a beatmap in the library just by typing it out, but what you're looking for right now is the "Tutorial". Since this is your first-time, the only beatmap inside is "Peter Lambert - osu! tutorial". The beatmap will teach you about the fundamental of osu! Standard play such as Hit-circles, Sliders and Spinners. For playing techniques, you have to learn as you go, as they can't be taught verbally.
How to start playing your favourite songs!
Step 1 - Find the song!
What is a beatmap you might ask? Just like any game, it will have levels, or sometimes maps. Well in osu!, we have "Beatmaps" which are exactly what you think they are, they are maps with beat (rhythm). Finding beatmaps for songs is easy! You can find every submitted beatmap here. You can type anything related to your search: song's name, artist's name, album's name, even the genre! For this example we'll try find "Coldplay - Don't Let It Break Your Heart"

Step 2 - Download it!
If there is an approved beatmap version of the song you are looking for, you should find it in the result pages. However don't be upset if your song isn't there! There are hundreds of beatmaps being submitted and reviewed so your song might be on its' way! Please do not post requests for beatmaps you'd want, most mappers will map the songs they love, not what you love, but that might be a hint for you to start your own mapping career!
Once on the beatmap's page, there should be a giant "Download Beatmap" button (you have to be signed in first, otherwise, a yellow button will direct you to sign-in).

Note: It doesn't matter where you save the file as long as you can find it once the download is finished. Also, you can choose to directly open it but will force your osu! to start-up if it's not activated.
Step 3 - Load it up with osu!
Once your download has finished, you should simply be able to open the beatmap .osz file. If for some reason that doesn't suit you, you might want to hold & drag the file into the osu! window.

Step 4 - Play for fun or compete with other players!
Once opened, osu! should start-up. To start browsing your beatmap library just click the massive osu! circle in the middle, click "Play", "Solo". Your new beatmap (along with the tutorial) should be listed there. Choose the difficulty you want to play and you're off to become an osu! player! You may also play "Multi" instead of "Solo" where players will compete in live games. You can also be part of the fun.

There we go! Now you're ready to play osu! and find your favorite songs. If you're having any issues, feel free to ask for help in the forums or in-game (press F8 to open chat window, under #osu).
GOOD LUCK AND HAVE FUN! :3
Game Modes
Solo
So what exactly is the "Solo" game mode? Basically, this is the game mode that can be played both offline and online by yourself. This game mode has no limits, you can play any map you have, use any Game Modifiers (sometimes called "score modifiers") you want, pause the game when your hand gets too sore, and restart it when ever you feel that your score or accuracy is too low. You can also check out the Local ("This" machine) or Online leaderboard and see the results or watch the replays simply by clicking their tab. You can press "F2" to save a replay externally so you can enjoy the replay without the hassle of searching for it.
Multiplayer
If you feel a bit lonely playing alone in "Solo" mode, you can always join the multiplayer lobby. The minimum requirement to enter the multiplayer lobby is only an osu! account logged into the game with a decent Internet connection. To gain access, press the "Multi" button. There, you will see heaps of rooms ranging from Easy to Insane levels. Playing in multiplayer is a really good way to make new friends and compare your skill with other players. Once you join a room, you will probably have to download the beatmap chosen by the host. Click on "Click here to download this map" (should be on the right side, in painfully red tab), and your web browser should open up the selected beatmap's listing page. Just click on the big "Download Beatmap" button (you need to sign in first though) and open the .osz file as soon as the download finishes! Don't worry if you think your download is too slow, most hosts will wait for your download to finish and they will ask you for the percentage of the download. Don't be shy and tell them! If you feel it may take a long while, you can also tell them to play without you (Host has a special command for it, Force Start Game). The multiplayer gameplay is slightly different from solo, as you will not be able to pause the game and mods are limited for players other than Host (Host have full privilege and can give permission by enabling "Free Mods"). The most unique feature about Multiplay is Tag Co-Op (or Team Co-Op for team type) and Team VS. Also, you can see your opponents/team-mates' scores as you play, creating an intense atmosphere as you try to secure the top spot.
Competitiveness
This may look like a simple game of clicking circles and holding sliders, but it can get really intense and serious! Here's a few things you should know about osu! rank system.
Levels
The player's level only represents the amount of score (ranked or unranked) the player has gained regardless of finishing the beatmap or not. In short, relative experience of the player in the game. However, unlike other games (read: MMO games) you might of played, osu! does NOT give any advantages to any players whatsoever. A naturally gifted level 10 player can potentially beat a level 100 player (this means that all the level says is that the level 100 player spent more time playing beatmaps online). Also, keep in mind that you get higher scores when you get higher combos (hitting notes without missing any in a row).
Performance Points
Performance Points are basically points used to measure the player's competitiveness. The more points you get, the better the rank you have comparing relatively among other online players. There are a lot of different calculations involved to count everyone's points, but keep one thing in mind; you get pp when you reach a high rank in any ranked map. In most cases, to reach a considerable top rank, you will need to full-combo a song (don't miss a single note) to get the highest score (from the combo multiplier). A player with 99% accuracy that missed one note somewhere in the middle of the song can be outscored by a player with 80% accuracy who didn't miss a single note. The score deviation (eg: 2.8 mil vs 5.0 mil) will be pretty obvious when the song is long (2 ~ 5 mins). However, you might have to apply some difficulty increasing score modifiers (like HR,HD) to get an even higher score!
Accuracy
This measurement is the average accuracy among all the ranked scores the player played. It is split into two types, player's overall and pp's accuracy. In your player tab in-game, the accuracy shown is pp's accuracy. For player's overall, you may have to visit the "Online Results" after you played a beatmap online or refer to your own forum page. In osu!, accuracy measures not only your aim, but the timing as well, so it is important to tap the notes perfectly to the beat. Pay extra attention to the approach circles!
Achievements
Achievements are little badges on one's profile which indicate that someone has accomplished an exceptional goal. Achievements' badge can be unlocked in various ways such as playing a lot or hitting a certain combo count in one beatmap. When you fulfilled the criteria, you will receive a badge at "Online Results" and your account's personal achievement tab. However, the meat of it is the hidden mission, Hush-Hush, where the criteria is not known and it is up to us to figure it out on how to get it. They are pretty hard to get but it is fulfilling when you managed to achieved it by yourself.
Note: Achievement badges can only be unlocked online based on your current account logged-in.
Chat
Knowing the basics on how the Chat Console works is important to those who are interested into making new friends in our friendly environment. It's also important to understand how to use it in case someone messages you (unlikely to happen unless you are well-known), telling you tips or asking for help themselves. There is a button to open up the chat on the bottom right of the screen but it's way quicker to just press F8 on your keyboard. You will have a few tabs like "#osu!", "#announce", "#userlog" or if you're in a conversation with someone, you'll see their name as a separate tab. Navigating through the tabs is extremely easy, just press on the tab and type your message then press "ENTER" to send the message to the opened tab. Regardless, please do not spam (continuously sending short messages into the chat) as doing so will automatically mute your account for a few minutes and your "hard-work" removed from the chat. You can press "F9" to open the user list (called Extended Chat Console) connected to the server, and you can apply filter to narrow down your search.

Those are the basics, don't be shy to ask for help in-game or in forums, also the osu!wiki is here to help understand the game more.
Beatmapping (Edit)
osu! offers an opportunity to any player to create their own beatmaps. If you think you're creative and passionate about learning, you may start mapping your own songs. To start, simply access the "Edit" mode and drag your song's .mp3 file onto the editor. Press "F5" to refresh the song list and you should find a blank map with your song name in it. However, beatmapping is a very complicated process which requires a bit of dedication, and thus it will not be covered on this page. The best way to learn is to try it out yourself. Also, "Edit" mode can also be useful to take a closer look at any beatmap so you can see whats up with that part where you always fail! Also try to avoid editing ranked maps because they will automatically revert to being unranked (locally only of course) and you will not be able to earn pp/accuracy from it until you update the map again.