Cards from all regular core sets and expansions since Eighth Edition are legal.
Timeshifted cards in Time Spiral (even never reprinted in the modern frame), cards from Planeswalker decks, and Buy-a-Box promos are considered legal in this format. However, cards reprinted as Masterpiece Series are not considered as a part of the respective expansion, and can only be used if they are already legal in Modern.
Modern Horizons (2019) introduces new cards into Modern without them ever being legal in Standard. It will contain new cards and reprints which are currently not legal in Modern.
The Modern Masters series contains reprints of Modern-legal cards from Eighth Edition onward. The sets include many cards with new art as well as new, never-before-printed tokens.
As of January 25, 2019 the following cards are banned in Modern tournaments:
Constructed decks must contain a minimum of sixty cards. There is no maximum deck size; however, you must be able to shuffle your deck with no assistance. If a player wishes to use a sideboard, it must have fifteen or fewer cards.
With the exception of basic land cards, a player’s combined deck and sideboard may not contain more than four of any individual card, counted by its English card title equivalent. All cards named Plains, Island, Swamp, Mountain, and Forest are basic. (The five Ice Age block snow-covered lands -- Snow-Covered Plains, Snow-Covered Island, Snow-Covered Swamp, Snow-Covered Mountain, and Snow-Covered Forest -- are also basic lands. Note that snow-covered lands are permitted only in formats that allow expansions from the Ice Age block to be used.)
Official information about Constructed formats and the Modern format can be found in the Magic: The Gathering Tournament Rules.