In the new Terraria 1.4+ "Journey's End" Update several new game difficulties were added.
These game modes have to be set during world generation process, and become a permanent feature of that world. Normally they cannot be changed after world generation. This guide will help you understand what the difference is between these game difficulties.
There are editors that can be used on game save to change which difficulty it was created with. After the 1.4+ update, we've seen some scattered reports that attempts to use these editors resulted in some corruption of game saves, so it might be necessary to wait for the authors of these editors to update them to work correctly with 1.4+ version of Terraria.
This is a new difficulty added in the 1.4+ update, and offers much more relaxed and casual gameplay. Characters start with a whole entire package of gears (weapons, tools, supplies, etc) and using any of this gear is optional, so you decide how easy you want to make it (or not). Journey mode also adds a unique feature, where you can research blocks and items, and then duplicate them whenever you want! You can make entire stacks of blocks to build whatever you want, or defeat a boss once and then just duplicate the special loot so everyone gets a copy of a weapon or item from the boss loot! It's sort of like aspects of creative mode, but still in a survival game. There are still monsters, and you can still take damage.
This is the standard gameplay experience that people have enjoyed for years. Your character will start with only a few basic tools and weapons (shortsword, axe, and pick), and monsters do normal amounts of damage, and have normal amounts of health. Boss fights are challenging, but possible to defeat with the right weapons and gear.
Expert mode offers much more challenging gameplay. All monsters do more damage, and have more health. Bosses also have more health (a lot more!), and have new unique attacks and fight patterns that are a lot harder.
When you beat a boss however, you receive a special treasure bag from them; those treasure bags have unique boss loot you only ever get in expert mode .One of those special boss items (called the "demon heart") grants your character an extra accessory slot when you use it, giving you a 6th accessory slot! This extra slot is only usable in expert or master mode worlds - if you go to a Classic or Journey Mode world, that slot is grayed out and the item in the slot won't have any effect.
This is also a new difficulty mode added in the 1.4+ "Journey's End" update. This has all the content from Expert Mode, but with even more challenges!
Monsters do even more damage, and have even more hit points than in Expert Mode. This basically gives monsters 3 times their normal health. For example, blue slimes have 75 health, instead of 25 health in Classic Mode. When you enter Hard Mode (after defeating the Wall of Flesh boss), the health of monsters increases dramatically in Master Mode. That same blue slime will have 411 health!
Characters immediately start with an extra 6th accessory slot in a Master Mode world, which can be combined with the extra slot gained from the Demon Heart, to make a total of 7 accessory slots. Bosses still drop the treasure bags from Expert Mode, but they also drop special "Relic" Statues of the boss, and special Master Mode only pets too!
You may see references to a "Hard Mode" in information about the game Terraria. This isn't a game difficulty mode you can set during world generation, but a special event that happens in any difficulty of world.
You "flip" a world to hard mode when you find and defeat the "Wall of Flesh" boss monster for the first time. This makes a permanent change in any difficulty mode world, where much more difficult monsters begin appearing, and some very drastic changes are made in the world, that continue as you find and defeat further bosses. Even in the Classic Mode, if you flip the world to Hard Mode when you aren't prepared, you will die. A lot. In Expert and Master Modes, this change is even more severe.
When a world enters "Hard Mode", a whole new biome called the "Hallow" appears in a giant diagonal stripe from where you killed the Wall of Flesh boss, reaching from the Hell layer all the way up to the sky. Any blocks in the way are instantly converted to Hallowed blocks. The Hallow spawns "Light" themed monsters, made of pure good. That doesn't mean they're friendly, they're quite hostile. While they're made of pure 110% goodness, you the player are not, so Hallow monsters consider anything not pure good to be basically... pure evil. A matching mirror image stripe of Corruption/Crimson appears (depending on what is normal in that world), which also instantly converts blocks to evil corruption/crimson. They will roughly create a V-shaped design. After a world enters Hard Mode, any hallow or corruption/crimson begins spreading a converting blocks at a much faster rate (and only returns to normal by finding and defeating bosses that appear later on).
More changes can occur by performing other actions, such as new ores being spawned, and other changes can happen as more boss monsters are found and defeated. This is the start of the main progression of the game, where defeating more bosses makes further changes in the world.