Depending on how forcefully the chair went over your toe and the weight of the chair, there is a possibility of a break. It will hurt whether there is severe bruising or any other damage. A young person I knew broke her toe when a small pack of frozen peas fell out of the freezer on to her toe.
The hospital just strapped one toe to the next one.
Like CP3kids mentions, apply a cold compress to the toe. I also that applying cold natural set yogurt (lots of it) and lightly strapping it will also help to draw out any inflammation.
Keep the foot elevated (raised) up as much as you can, to allow it to heal.
The light pink colour under the nail is a sign that there is bruising and this is likely over the next few days to turn darker or turn black. This will grow out as your nail grows which is quite slow. It depends how much damage if any was done to the nail bed with regard to whether the nail lifts off or not.
If after applying the cold compresses, the toe does not settle down, make an appointment to get it checked out.
I don't think that a chair rolling over your toe is heavy enough to cause you to loose the nail. Just keep watching it, but I am sure all will be OK.
You most likely have some bruised soft tissue and perhaps even chipped the bone in it. If the nail bed starts to become discolored and loosens, you may lose the nail. I think that when a nail avulsed (gets lifted up and away from point of origin) that it's more likely to happen with blunt force trauma like with a hammer and the ensuing swelling, or something being forced under it (like stubbing it on something that has a ledge on it). If it's swollen, you may want to try soaking it in some cold water, or applying some ice to it without the ice directly touching your skin ( in other words place something a washcloth or a piece of gauze over/around the ice, and then place that on your nail and skin. With the wheel of the chair rolling over your toe as the mechanism of injury, you will most likely not have an avulsion occur. Either take some Tylenol (acetaminophen) or Alleve ( Ibuprofen) to help with any pain, unless you are allergic or able to withstand any pain that ensues. Good luck!