I'm going to preface this with a disclaimer. I am not quoting any authority on this. I've done no research on the literature regarding the psychology of grieving. This is based on my observations and thoughts.
Americans don't know how to grieve (and here I'm talking about mourning the death of a loved one). So we shroud it in mystery, avoid talking and thinking about it, and muddle through with largely unhelpful behaviors. We have no idea how to help someone else grieve any more than we know how to do it ourselves.
I've had to deal with the death of 3 grandparents and this year two people that I worked closely with died (one expected, one not). I've also helped my partner through his father's death in 2001, and several friends through the death of their pets. I'm no stranger to grieving.
But the process of grieving is not magical or unknowable. It is merely painful. Here it is in a nutshell:
Grieving is the process of
1. Respecting the past
2. Accepting the present
3. Moving forward to the future
Let me walk you through these steps.
First, it is very important to respect the past. Remember the times you spent with the loved one, good and bad. They are all part of your memories. This does not mean you idealize the person/pet who has died. The biggest mistake people make is unloading a bunch of lies about how so-and-so was the "the nicest person" when they were actually a jerk. It is not helpful to lie about the dead. It is also not respectful. And it does not help people move on.
Second, it is important to accept that the loved one is dead. I believe in ripping the band-aid off quickly rather than prolonging the pain. It is the same with grieving. Candy-coating reality by saying someone has "moved on" or "is in a better place"* actually makes it harder for people to accept that that loved one is dead. Why would you want to prolong someone's agony by offering nice-sounding euphemisms just so you don't have to face the harsh reality that the grieving person is facing. All these euphemisms for death are is a way to psychologically distance yourself from the reality of death and the pain of loss. If you care about the person, be present in the face of their loss. Say things concretely and lovingly. Face the death of their loved one with them and they will accept it much faster and easier. Another helpful thing you can do is go out of your way to help the grieving person with everyday chores and other tasks. People tend to say "call me if you need anything." Instead, say "I'm coming over with food tonight so you don't have to cook. Is 7:00 okay?" And be there to listen to them.
*For the record, I've seen no definitive proof of a soul or an afterlife. I accept death as the end of a life and a return of that energy and matter back to the aggregate system known as nature. That is beautiful enough for me.
Finally, once acceptance of the present is achieved, you can look toward the future and living a life without the dead loved one. This does not mean that all is peachy-keen. No. You can expect to be reminded of the loved one in the littlest things and re-experience the feelings of loss rather acutely for some time. But You are able to continue functioning. Time is the key to this stage. In time, the acuteness lessens until it is a dull throb. Nostalgia sets in. And we move forward.
So now you know why I cringe when I go to creepy funerals where they talk vaguely about someone "moving on" as if they went into the next room, and only say nice-sounding things about the dead, as if they are afraid that if they tell the truth, the dead will haunt them. This is childish, and not helpful in any way.