'Tanking' is a general term for surviving an attempt to put damage on your ship. There are many kinds of tanks and techniques for tanking damage in Eve, however in general we can classify them under three broad categories - active tanking, buffer tanking and avoidance tanking. Both active and buffer tanking can also be broken down into shield, armour and hull tanking, depending on the component of your ship's HP that your tank focuses on.
Active tanking requires that the ship actively repair the damage it takes via the use of shield boosters or armour repairers. These are usually complimented by hardeners and in the case of shields, boost amplifiers to increase the effective amount of shield or armour that you're able to repair.
Active tanking is a very cap-intensive task, with without some way of sustaining your capacitor active tanking will quickly leave you dead in the water. In PVE this often means using capacitor recharge modules of some kind to make your ship 'cap stable' (i.e. capable of running its repair modules forever). In PVP this is not really viable, since it uses up too many slots and is easily ruined by energy neutralisers. Instead, active tanks in PVP are generally assisted by either a nosferatu (for frigate sized ships) or a cap booster (generally cruisers and larger) instead of cap regen modules. These allow you to inject energy into your capacitor regardless of neutralisation, and keep those cap-hungry modules running.
The great thing about an active tank is that if your repair capability is strong enough, you can last forever (or at least until you run out of cap). However to be effective, an active tank relies on you being able to repair damage as fast as (or nearly as fast as) you're receiving it. Once incoming damage is significantly higher than the amount you're able to repair, active tanks quickly become fairly useless since you simply can't get enough cycles in to make a difference.
Active tanks are especially effective on ships with a bonus to repair amount, such a myrmidons and cyclones. They work best in fairly small engagements, particularly solo, since in these cases the amount if incoming damage tends to be fairly manageable. Combat boosters (drugs) and command bonuses also help to make active tanks really shine. As a general guide, you should avoid using active tanks where the amount you're able to repair is not significant relative to the damage your ship is capable of dealing itself (for example, a 100 dps active tank on a cruiser is probably not worth it).
Whether shield or armor tanking, managing your capacitor with an active tank is paramount. You should experiment with how long you can leave your tank on without capping out, and keep a close eye on your capacitor during combat. In a fleet it is often better to reduce damage output to maintain tank if under fire (e.g. pull range on your opponent to allow your tank to catch up) - if you're lucky they will keep trying to take you down giving your fleetmates the opportunity to kill them as they do so.
Buffer tanking emphasizes maximizing effective HP (often called EHP, this is the combination of raw HP and resists) in order to increase the amount of time it takes for your opponent to destroy your ship. This is done by strapping on the largest size shield extenders or armour plates you can fit, and backing them up with the associated hardeners to increase your resists.
Buffer tanks tend to be easier to fit than (effective) active tanks, and are generally more popular for PVP. They are effective on almost any ship, and generally outperform active tanks once the engagement is larger than a handful of ships on each side. Buffer tanks also combine well with logistics ships thanks to their typically high resists.
To demonstrate the comparison between active and buffer tanking, let's say our active tanked battlecruiser is capable of tanking 500 dps and has 25,000 EHP, while our buffer tanked battlecruiser has 50,000 EHP but no repair. In order for the active tank to be more effective, it needs to repair at least 25,000 EHP before it dies, which will take 50 seconds. This means that against any opposing force dealing 1000 dps or more (for reference, this is roughly 2 battlecruisers or 3 cruisers) the buffer tank will be more effective, while against any force dealing less than that amount, the active tank will be more effective.
Obviously the above example assumes enemy dps remains constant and ignores factors like enemy ships being destroyed as the fight progresses, however it should provide a decent illustration of how the two tanking strategies compare.
Another strategy related to buffer tanking is Passive Regen tanking - this is exclusive to shield tankers, and involves increasing shield hitpoints as well as shield recharge rate. Thanks to the way shield recharge time remains the same regardless of total shield HP, this can be used to produce a similar effect to a strong active tank, without requiring an actual repair module and while maintaining a fairly significant buffer tank too. However, Passive Regen tanks typically take up both your mid and low slots to be truly effective, and are not recommended for PVP outside of very specific circumstances.
The final technique in tanking is avoiding damage. This actually covers a variety of techniques and goes by many names including 'speed tanking', 'sig tanking', 'ewar tanking', 'kiting' or 'getting under their guns'. The key principle they have in common it putting yourself in a situation where it's difficult for your opponent to deal their full damage against you.
On speed tanking, it's important to note that simply going fast doesn't reduce the damage you take - especially if that speed is due to a microwarpdrive which drastically increases your signature radius. What it does do is allow you to get quickly into a position where damage is reduced, either by increasing the range from your target or by quickly closing to a range where (once your mwd is turned off) your target will be unable to track you effectively. True 'speed tanking' in a form that actually reduces damage also requires you to keep a low signature radius, which means using an afterburner.
Shield, Armour or Structure Tanking
Both armour and shield tanks are popular in PVP these days. When buffer tanking, armour tanking typically provides greater effective HP (the combination of raw HP and resists) at the expense speed and agility, and limit the low slots you have available to enhance your damage output, speed, or range. Shield tanking on the other hand provides somewhat lower EHP and causes an increase to signature radius and limits the mid slots you have available for tackle and propulsion mods, but leaves your low slots free.
For active tanks, an armour tank typically provides more repair per unit of cap, however each individual repair module is less effective. It is not uncommon for active armour tankers to rely on two or more armour repairers where a shield tanker would only require a single booster. The repair modules themselves don't confer any penalties, however the associated rigs share the penalties of the passive tanking modules.
Most ships can either be used with a shield or armour tank, although many will favour one over the other. The key factors are bonuses (if they exist) and slot layout - since a propulsion module and often at least one tackle or ewar mod are usually necessary, fitting a shield tank on a ship with few mid slots can be especially difficult. This is even more true when it comes to active tanks, since you'll usually be adding a cap booster to the mix!
Structure tanking is rare and generally not effective. Active hull repairers have far too little effect for a reasonable active tank, however passive hull tanking can be both interesting and unexpected. Gallente ships are especially good at structure tanking due to their large amount of structure hitpoints. With only a damage control, some Gallente ships will find the largest part of their effective hit points are in structure! That said, a true structure tank is a rare thing and generally involved modules like reinforced bulkheads to increase structure hitpoints. These tanks generally aren't viable in comparison to a shield or armour tank, however they can be very useful for baiting (a Gallente battleship can fit a huge structure tank, and as their shields and armour melt away quickly, the enemy will often keep them primary even though there may be greater threats on the field.
Choosing a Tanking Strategy
Which tanking strategy you choose will depend on your gang, your ship, and your expected targets. Consider your bonuses and slot layout - be cautious of fitting purely active tanks on a ship with no resist or repair bonuses (unless it's a frigate), however that doesn't mean it's always a bad idea. If your gang includes logistics ships, make sure your tank is compatible with them - NEVER bring an armour tanked ship to a shield logistics gang or vice versa, unless you have a very good reason for doing so.
Mixing Tanking Strategies
Sometimes, combining one or more tanking strategies can be very effective. Combining avoidance tanking with either buffer or active repair can work very well, effectively reducing the amount of damage you need to repair or prolonging the life of your buffer. Similarly combining an amount of active repair with a buffer tank can work well in many situations, although you should be wary of significantly reducing your buffer in order to do so (obviously combining active repair from logistics ships with strong buffer tanks is an excellent and very popular strategy).