How can I know whether my Apple Care is registered?

It's automatically registered when you purchase it as per this support article:

http://www.apple.com/uk/support/products/faqs.html

As the article advises it can take up to 24 hours before it goes through. Give it a day or two and then to check if it is registered, make a note your device's serial number (look at the box it came in which you should still have) and type it into the field here:

https://checkcoverage.apple.com/gb/en;jsessionid=r1swWS5X3x6GWrLhfhVt6BfGM5TSRdq 678mPhCcy8QpcL85cWVhM!246325129?locale=e…

The Repairs & Service Coverage Section should have an expiration date. You should be able to work it out from there? If you purchased AppleCare afterwards but within the first year standard warranty then it too would have been automatically registered.

It is registered IF you buy it in an Apple store along with a computer or iDevice. It is NOT automatically registered if you buy it elsewhere such as an online reseller, ebay, amazon, etc. You receive a box with a small booklet in it showing the legal terms and, there is an insert with a very, very long registration number. Go to your support profile and choose to register your Applecare by choosing which device it is for and entering the registration number.

FWIW, it must be registered within 364 days of the date of the computer purchase (I believe 90 days for an iDevice) or it is not valid.

Register your AppleCare plan - Apple Support

Also note that the legal terms may vary from continent/country as they do for the regular warranty - I posted a link from Apple's US site; the previous links are from the UK site.