Combat is a hybrid of what you would see in any other Elder Scrolls game and the hotbar-based abilities of your standard MMO. Though the game takes place from the first-person perspective by default, It's a little easier to manage combat in third-person. Elder Scrolls' standard resource bars are present (magicka, stamina), and using abilities will draw from the appropriate pool. Mashing on the left mouse button will use your standard attacks, and you can trigger power attacks by holding it down for longer and moving in a certain direction. The right mouse button provides a block, and if you trigger an attack while blocking, you can bash the enemy to knock it out of spellcasting or its own power attack.
Certain abilities also trigger synergies. While in range of another player who uses an ability that has synergy, you'll be prompted to interact with it, often boosting its power or providing you with a short term buff. In group play, one of the valuable abilities is Blood Funnel, which drops an altar that players can click on and get a boost to their health regeneration. The more players that click it, the more powerful the effect for everybody. My Nightblade has an ultimate ability called Consuming Darkness, which snares enemies in an area and reduces my damage taken. If other players are around, they can interact with it to gain a temporary invisibility and escape combat. Synergies are extremely important, and mastering how to use them is key in defeating most of the group content I encountered.
The three Alliances are also locked in a war to control Cyrodil, the Imperial seat. This is the PVP portion of ESO, called The Alliance War. Once you hit level 10, this option becomes available to players, though I must admit that I haven't put much time into it for this writeup.
Like other Elder Scrolls games, there are dungeons, ruins and caves all over the place. You might expect these to be instanced content, but they're open to anybody who wants to explore. Sometimes they culminate in a miniboss that is soloable, but is best experienced with friends. Luckily, even if you're not grouped, all of those synergies I described above are available. A courteous player will often use synergies to help strangers along.
Certain areas of the world contain powerful monsters or small events that are intended to be taken on as a group. These are similar to the public quests you might see in a game like WAR or Guild Wars 2, but a little smaller in scale. Completing them will give you some nice rewards, but if you see a skull with crossed swords on your minimap, make sure you bring friends.