Can I levitate an object without using an electromagnet? I know that it's possible to make an object levitate using an electromagnet to hold it up. 
But is it also possible to do this with regular magnets? Is there a special kind of magnet I need in order to have one powerful enough to hold an object up?
I'm asking because I have this idea in mind where I want to make a decorative item that is levitating an inch or so above its container.  

A is the object I want to levitate.
B is the container.
C shows the magnetic field that is repelling A and B from each other achieving the levitation.  
The size of this would be as small as a food plate, maybe even smaller. B is barely a pound or two in weight.  
Is that possible?  
 A: Earnshaw's theorem proves that using only static ferromagnetism (like refridgerator magnets, which have a very slowly changing magnetic field) it is impossible to stably levitate against gravity.
http://en.wikipedia.org/wiki/Earnshaw%27s_theorem.
Diamagnetic materials might be used to levitate objects.
A: The references in other answers to the Wikipedia pages about Earnshaw's theorem and Magnetic levitation are right on; you absolutely cannot have stable, static magnetic levitation using only ferromagnets. However, at the time of my posting, both other answers contain misinformation.
Kakemonsteret's answer is incorrect*, because Earnshaw's theorem is not the end of the story. If you violate the assumptions of Earnshaw's theorem, its conclusions are no longer valid. There is a special kind of magnet Earnshaw's theorem didn't consider, which you can use for stable, static magnetic levitation: a diamagnet.
Superconductors are strongly diamagnetic, but I suspect using a superconductor is an unsatisfactory solution, unless you have liquid nitrogen on tap. Weaker room-temperature diamagnets like bismuth and pyrolytic carbon don't support much weight, but they will levitate forever and don't require cooling, electricity, or attention.
user879's answer links to the excellent Wikipedia page on magnetic levitation. I'd like to emphasize that this page explicitly discusses diamagnetic levitation, which I think is the closest answer to the spirit of your question: a special type of magnet that makes an object levitate without using an electromagnet. However, the answer goes on to state that a bowl of [ferro]magnets would also work. This is false.
I spent a full summer in high school trying to float a tennis ball covered in fridge magnets above a bowl lined with fridge magnets. I failed; I wasn't violating any of Earnshaw's assumptions. Years later when I finally learned Earnshaw's theorem, I was relieved to realize my failure was predictable, since my goal was provably impossible.
So, buy some strong ferromagnets, buy some strong, light diamagnets, scale down your design's ambitions to something very light, and start building your decorative item.
Don't put your credit cards or your hard drives near it.
*EDIT: This answer has been updated, and now mentions diamagnets.
EDIT: I can't believe I forgot to mention my favorite aspect of Earnshaw's theorem: it proves there's no such thing as solid objects! This is the single most obvious everyday evidence for quantum mechanics. If you see an apparently static, stable object that isn't held together by gravity, then it can't be modeled as a collection of point charges interacting through Newton's laws and Maxwell's equations.
A: Answer is NO.
I would summarise the question as below.
'Is it possible to make a system with 2 parts made of permanent magnets so as to have an arrangement of one part floating due to magnetic repulsion of the other part above it ( effectively the gravitational pull gets balanced by the magnetic repulsive force )
As we know the likes are repulsive and unlikes are attractive in magnetism, 
Suppose the north pole (upside) of the magnet placed below (base magnet) is repulsing with the north pole of magnet (floating magnet) which is placed directly above the base magnet with its north pole facing down. Then during this time, the upside of the Floating magnet is attracted by upside of the base magnet. 
This combination of repulsion and attraction forces creates a couple in the Floating magnet(unless it is hinged some where without any physical contact with the surroundings or base magnet)
Due to this force the Floating magnet will rotate and stick on to base magnet immediately with its southbpole mating with north pole of base magnet having north pole as upside.
Hence this cannot be used as a decorative item.
But yes, it can be used as a decorative item with Floating magnet if the one magnet is a controllable ectromagnet which can take care of the magnetic strength for balancing of the Floating magnet.
Hope this clarifies the query.
A: yes, you can.  Not easily though.
My favorite is the Meisner effect.  It would require a superconductor for B, which would most likely require liquid nitrogen.  Basically, the levitating magnet (A) would create an current in the superconductor (which has "zero" resistance).  The current would then go for as long as the superconductor superconducts, creating a magnetic field, which would repulse A.
For more explanation, check out wikipedia.  Basically, a  I have no idea how much this would cost.  Although, the liquid nitrogen needed would most likely make it impractical.  By the way, there are no room temp superconductors currently known.
more methods are shown here: http://en.wikipedia.org/wiki/Magnetic_levitation.  Some methods shown there do in fact use electricity, but some don't.  Check out the spin-stabilized levitation.  Very cool.
The last would be to build a bowl of magnets.  If you arrange it right, you should be able to get it stable.
