Today a zoologist told me that prawns and fish are very similar, but zoologists/taxonomists have put them in different categories because their distinction is unclear (like hydrogen in the periodic table). This completely bewildered me, as I know prawn and fish are in different classes because there is no similarity between them. I have no connection with biology now but studied it in high school — but even then, I could not believe it. Can someone please elaborate on this?
Introduction to phylogeny
What makes that two species being closely related or not has nothing to do with whether they look a like or whether they live in similar environment. It has to do with their evolutionary history. Evolutionary history used to be inferred from phenotypic traits ('phenotype'≈'how an individual looks like') but today it is most often inferred from genetic data.
First you should have a look at this recent answer to get a short introduction to phylogeny.
Phylogeny of animals
Here is a basic tree of all animals
A fish is any member of a group of organisms that consist of all gill-bearing aquatic craniate animals.
Craniates are Chordates, therefore fishes are chordates. You can have a look at the position of chordates on the above graph. Note that all mammals, birds, reptiles and others are also chordates. Note that fishes do not represent a monophyletic group (if you don't understand the term 'monophyletic', then you need to read the linked post above).
A prawn is a crustacean. Crustacean (incl. prawns, shrimp, crabs, ...) and Hexapoda together form a monophyletic group. Crustaceans are arthropods and are therefore much more closely related to insects, molluscs and spiders than to any lineage that we would call fish. Here is a tree of arthropods (and related clades). See the position of arthropods in the above tree. You can also have a look at the position of crustaceans within arthropods below. Please note that the tree is a bit outdated a present crustaceans as being monophyletic which is wrong (according @har-wradim, see comment below).
Your discussion with your friend
There is no doubt about the phylogenetic position of crustaceans as being part of arthropods and therefore not being very closely related to fishes. I think you misunderstood what your friend tried to tell you.
A note on developmental biology
All multicellular animals, undergo a grastrulation during development during whihc a hole is formed called the blastopore. In all deuterostomia (incl. fishes) the blastopore becomes the anus, while in prostomia (sister clade to deuterosomia which includes arthropods), the blastopore becomes the mouth. This a very major difference in how these different clades develop from a single cell (after fertilization) to a multicellular adult.
If you want to have a look at the tree of life, you can find great ressources on this post.