Generally, the only items you need to worry about that can only be used by certain classes are scrolls, wands and staves. I think some rods may also fall into this category.
In this situation, a character has to have the spell on his spell list to be able to cast the spell, and there is also a requirement that his spell-casting attribute be high enough to cast that spell... but that may only apply to scrolls. In the case of scrolls, if he is unable to cast the spell at his current level, he needs to make a caster level check.
For these types of items, the only way you can use them if the spell is NOT on you spell list is the Use Magic Device skill.
Magic Weapons (even Spell Storing magic weapons charged with a spell by another user), wonderous items, potions, rings, etc... can all be used by anyone unless the item specifies differently.
If I remember, the rules on this are in the DMG near the beginning of the Magic Items section. Scroll use is detailed in the scroll generator section.