Firstly decide, if you buy a wheelchair privately, that it meets your needs better than a wheelchair you may have been supplied with from your local Hospital Wheelchair Service (assuming you have been referred for one by your GP etc).
Is it for transport round the home, school, workplace, outdoors, indoors, on different floors and does the wheelchair need to be transported in a vehicle.
If the wheelchair needs transporting in a car or other vehicle it will need to fold up easily. Some whelchairs have detachable parts for this purpose. The person putting the wheelchair into a car boot for example should be able to manage it easily
Folding wheelchairs are usually heavier than rigid ones but can give a smoother ride over bumpy surfaces because they are less rigid and "give" more.
If the wheelchair user has poor posture the wheelchair must not exacerbate any medical condition the user has. If their are any doubts about this advice from a professional such as an Occupational Therapst should be sought.
People with good upper body strength who use the wheelchair for sport can purchase lightweight self-propel chairs with tight turning circles and maneouvrability.
Other people may require the stability of a wider based chair and if poor muscle tone is present belts to secure the person in the chair.
The main consideration here is the tyres.
Pneumatic tyres are best for all round use because they give the smoothest ride over bumps. Regular tyre pressure checking is required.
Solid tyres can be used if the main usage is indoors.
For moving over rough surfaces outdoors larger tyres with a greater surface area and pronounced tread aid a smoother ride.
Most chairs will have swing away legrests and fold up footplates. These aid independence for users capable of transferring themselves and attendants helping users who can't transfer themselves.
Some wheelchairs have removeable sides which enable transfer of the user to a car seat or toilet for example via a transfer board.
Sitting in a wheelchair for long periods may leave the user prone to pressure sores and postural discomfort. A wheelchair cushion can greatly reduce this risk and many types are available.
Finally there are many wheelchairs available for purchase and a clear idea of which one suits your requirements is very helpful.