If you want to look at it scientifically it has been proven that alloparenting (apoting and looking after another species) releases the same chemicals in the brain, such as oxytocin, that are involved with pair-bonding and maternal attachment.
Or if you prefer, we love our pets on the same level as we would love any other family member. When they die the grief is just as real and as deep as if we'd lost a human family member or friend. For some people the pet is their best friend, and may be the only other being that they speak to from day to day.
There are set stages to grief and you will go through them, shock, denial, guilt, anxiety, depression, anger etc. There are also cognitive, physical, social, spiritual and physiological aspects of grief. Grief is individual to each person and everyone will have to find their way though at their own pace.
I believe that the loss of a pet probably brings out the guilt aspect of grief more than any of the other emotion, the feeling of "have I done everything to save my pet's life" or for some people not being able to afford ongoing vet bills, the act of signing the slip of paper to allow the vet to euthanise the pet all contribute to this. Part of loving someone is knowing when to love them enough to let them go and leave their life of pain, although it will leave you heartbroken.
Don't neglect yourself after the loss of a pet, you will be emotionally fragile, think about how you would advise someone in your postition, what about holding a memorial service?
Do what you have to do to get through each day; don't let the guilt overtake you when you do something you enjoy. Treat yourself, maybe to a film or a box of chocolates, don't keep your feelings bottled up. Other family members and friends will also be feeling the loss, talk about your pet and about how you are coping, journal if you don't think you can cope with talking out loud. You may even want to talk to your GP about grief counselling or antidepressants to help you through the worst.
You may also have lost your routine, getting up in the morning to look after your pet, walking the dog after work, mealtimes and time spent together have all gone as well. Try to find a new routine so that you don't focus on the what you would have done.
Remember the intense feelings of immediate loss will lessen as time passes. One day you will look back on your memories and smile, know this is a sign of recovery and not forgetting. That is when you can start looking to the future, it may be time to give all that love to another pet, or to help a neighbour with theirs. You could volunteer at a rescue centre, you never know if that will lead to finding a new best friend.