Let’s make a short resume of how to equip the various hero items. The first thing that needs to be taken into consideration is the fact that there are essentially two main categories of items:

• Equipment
• Consumables

Both type of items need to be equipped on your hero first in order to work but the general difference between equipment items and consumables is that equipment can last indefinitely and it provides different bonuses to your hero, the troops that accompany him (be it on a raid/attack or in defense) and some even give you certain boosts similar to the ones certain artifacts give.

There is a number of equipment which will increase the hero’s regeneration rate, decrease the damage he receives in battle or decrease training times in stables/barracks, give additional culture point production, increase the damage the hero causes and boost the off/def values of the troops that accompany him. There is also equipment which shortens travel times, return times and improves general troop movement. For any of these items to work, they have to be equipped on your hero first – they will give no effect if they are simply stored in your hero’s backpack.
Since equipment represents such a vast number of items (all of which are already listed on the Answers here), it would be a really long list if I were to try and explain separately how to equip each item. But equipping items is not that difficult; all of them are already categorized both in the auctions and adventures as items which are meant for the left hand, right hand, shoes, helmets, etc. So, it is pretty self explanatory where these items need to be put in order to be equipped. The hero system will not allow you to put a right hand item anywhere else than in the right hand, so there is really no way to make a mistake.

As you can see in the picture above the effect of the item is shown in a mouseover if you go with the mouse over the respective item.
In any case, to equip any item in your hero’s storage, just click on it and drag it into its appropriate place on your hero. You will not be able to drag it into a wrong place on your hero - additionally the appropriate place will get highlighted at the moment you click on the respective equipment (you can see this on the picture above). Repeat the same action in reverse when you want to unequip it.

Now, consumables function a bit differently. First off, consumables are spent (consumed) – this means they give a one-time bonus or effect after which you need to get more in order to get their bonus again. Such typical items are for example bandages, ointments, scrolls, artworks,and tablets of law. The items I’ve just mentioned are also stackable, which means that they stack when you get more, so you may have e.g. 500 ointments occupying just one storage slot. Furthermore, there is no limit regarding the number of stackable items your hero can bring at a time. This means that there is no difference between bringing/equipping, for example, just one ointment or bringing/equipping 500 or more. Once these items have been used, the amount that has been used will disappear – which means that they have been consumed. Keep in mind that for all of these items to work – including the bucket, bandages, artwork, tablets, cages and so on – they need to be equipped first! Just having them in your hero’s storage won’t work and they won’t take effect.
There is only one place on your hero where all consumables go and that is the small box to your hero’s left (or, to your right) near his feet. You will not be able to drag it into a wrong place on your hero - additionally the appropriate place will get highlighted at the moment you click on the respective consumable:

So, remember: in order for cages, buckets, bandages and other such items to work, they need to be placed/equipped in this box first, prior to any battles or attacks, so they can be activated later at any time if you need them.