There are many types of magnetic materials. Objects commonly referred to as magnets are typically made of metals such as iron, nickel, cobalt, neodymium-iron-boron, and sometimes are made of minerals able to hold a magnetic charge. Magnets also exist as electromagnets, which have the potential to be the strongest magnets known. The strength of a magnet is primary determined by how it's made.
Certain minerals and metals can hold a magnetic charge. These are called permanent magnets because they hold their charge without needing electricity. They are not truly permanent because they can be demagnetized by being introduced to a much stronger magnetic field.
These magnets can be made by melting the component ingredients and cooling them in the presence of a magnetic field. The stronger the magnetic field, the stronger the resulting magnet up to the limit of the material. Neodymium-iron-boron (NIB) magnets are called rare-earth magnets and are the strongest permanent magnet we know of. Electromagnets use different materials and methods and should be discussed separately.
The bigger the magnet is, the stronger it will be. This even applies when comparing magnets made of different materials; a huge iron magnet can still be more powerful than a tiny NIB magnet. This applies to electromagnets as well. The more wire, and the thicker the wire, the higher potential for magnetic power that the magnet has. Of course there are still other determinants of an electromagnets power.
The power of a magnet is concentrated at its poles and is diminished on the sides. Very long skinny magnets that have their poles on each end would have very little pull in the middle of one of the sides of the magnet. Furthermore, for a strong magnet, contain the most magnetic force in the smallest volume possible. This means that if one magnet were rolled out thin and flat, like paper, while another of equal mass were formed into a cube, the cube would be the stronger magnet. Although total pull would be the same for both magnets, force at any one point would differ.
The absolute distance between a magnet and its object of attraction determines the amount of pull the the magnet exerts. A magnet covered in a thin layer of wood would have much less pull on a piece of iron than if the bare magnet were allowed to contact the iron directly.
With electromagnets, the strength of the magnet depends primarily upon how much current is flowing thought the conductor. More current makes for a stronger pull.
A straight conductor can still make an electromagnet if you pump enough electricity through it, but by winding a conductor around an iron core multiple times, you can create a stronger magnet that has a more useful polarity.