The first thing new players should know about Elder Scrolls Online is that combat is far different from any other MMORPG. It is probably closest to TERA Online (for the few people who have played that). For starters, weapons have no auto attack. Additionally, you can target enemies, you have to actually be aiming at them (with the crosshair in the center of the screen) for many of your single-target abilities and attacks to hit.

There are three types of attacks in ESO: regular attacks (classified as light or heavy), stamina-based attacks, magicka-based attacks. I will discuss the important aspects of each below:
Regular Attacks
Instead of auto-attacks, you have regular attacks. The damage dealt by these attacks seems to be dependent upon both weapon damage and stamina (even the Destruction Staff seems to use stamina for damage rather than magicka). These attacks cost no resources and can be used when you are drained of stamina or magicka. The range of these regular attacks depends on the equipped weapon (bows and staves will automatically attack with a ranged).

You can activate these attacks by left-clicking while aimed at the opponent you want to hit. If you tap click while being aimed at the opponent, you will hit them with a “light attack”. It takes about .75 seconds for this attack to execute regardless of the weapon equipped (can be made faster with enchants and weapon traits). If you hold down the left click, your character will charge up a slow attack over about a 1.5 second period before unleashing a stronger attack. This is known as a “heavy attack”.

Heavy attacks
Heavy attacks deal a little more than twice the damage of light attacks and take about twice as long to cast, so using Heavy Attacks will do slightly more DPS over purely light attacks. Heavy attacks will also knock down your opponent if they are off-balance. Heavy Attacks have the disadvantage of being interruptible (discussed below) and have a longer cast time.

Magicka-Based Attacks
Skills that burn Magicka are unlocked by putting points in your class skill tree. Magicka is also used by many guild and world skill trees as well. Magicka abilities can be used to deal damage, but they are often used to provide a number of utility effects, such as buffs, heals, stuns, or knockdowns.

No build (at least leveling builds) seem to be purely magicka based. Stamina seems to be too important when leveling as it is what powers your weapon skills. I am assuming at higher levels you get enough max magicka, magicka regeneration, and magicka recovery abilities to play a magicka-based playstyle. During the early levels you will need to rely at least partially on regular attacks or stamina-based abilities if you want to be effective.

Quick Note on Swapping Skills
After you have unlocked an ability via skill points, you can swap out your skills at any time. You can only have 5 active abilities an 1 Ultimate ability activated at a time, but you can equip and unequip skills at any time. At level 15, you can unlock weapon swapping, which will allow you to swap weapons even during combat. Each weapon can have its own unique abilities bar, allowing you to access an entirely new set of skills by pressing the weapon swap key.