Hi,
Below I have described what I did to set things up, underneath that in paragraph 2 is my fish's problem.

I bought 2 goldfish about a fortnight ago. They are both about 10cm long from tail to mouth. One of them had black bits on the end of his tail and bottom fins when I got him.
When I got them home I put them in the tank that I used to use when I kept goldfish a few years ago. (Of course, I washed it out in fresh water as it had been sitting around for ages). I put some new clean gravel stones in the bottom and some weed that was in their tank in the pet store. I also put in the filter I used to use to clean the water and provide oxygen. I put the Tap Water Conditioner Aqua Plus into the water.
Then I added my new fish.
The one with the black bits on appeared to be bullied by the other fish. He kept headbutting him and wouldn't let him get food. He was being really mean!
I rang the petstore and they said if one goldfish is stressed, the other one might take advantage of that. They said I could separate them. I didn't need to though because things calmed down.

WHAT IS WRONG NOW???:
Well now the one with the black markings on his tail and bottom fins seems really sick. He keeps sitting on top of the filter or on the bottom of the tank on the gravel. He lies sideways. He can swim around and get food but he doesn't do that unless I come along. If he isn't startled by a person being there, he just lies around on his side. He also has no top fin - I can't remember if he had one to start with! but I think he did. He hasn't got two of his little fins either. He has just two fins on the bottom near his gills and his tail (which is looking alot smaller than the other fish's). I have given him peas to try to make him upright. He can swim upright but when he stops, he has to use the weed to hold himself down and still turns sideways.
Is it me imagining things or could it be that his fins have disappeared! Has my other goldfish eaten them?!
I have moved him into a small plastic tub with some salt (1tsp per gallon as I read) and aqua plus.
Please tell me if I can save him. I expect to wake up in the morning and find him dead. It is very sad.
Also, is my other fish in danger of getting sick??!!!

I am sorry to hear of your sick fish.

First I would recommend that you answer all the questions at the top of the page. This is helpful to us when helping you.

Water quality is the utmost important thing in keeping gf. I am assuming that your tank is cycling which is very hard on fish and can kill them if not attended too.

You need to test for ammonia, nitrites, nitrates and PH to start with. You can get tests at the fish store. When you newly set up a tank testing every day is very important. Ammonia, nitrites can kill fish if they get too high.
This is what I am thinking is happening.
I will wait to comment further until you can post back.