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Usually I found the word "informal" beside the word in dictionaries.
e.g.
thanks exclamation INFORMAL: thank you noun Cambridge Dictionary
but sometimes I found the expression "non-standard form of ..."
e.g.
alright : non-standard form of all right. link
So, what is the difference between informal and non-standard form?
The term informal implies a relaxation of standards, which is not really the case with alright: it's just less common than the all right spelling.
Alright appears in the Oxford Dictionary, without any annotations about it being a non-standard form.
Merriam-Webster has this to say about it:
Although the spelling alright is nearly as old as all right, some critics have insisted alright is all wrong. Nevertheless it has its defenders and its users, who perhaps have been influenced by analogy with altogether and already. It is less frequent than all right but remains common especially in informal writing. It is quite common in fictional dialogue and is sometimes found in more formal writing (the first two years of medical school were alright — Gertrude Stein).
This NGram shows that it is nothing like as common as all right in publications, but that its usage has been growing steadily since the 1960's.
"Informal" would describe a usage which would not typically be found in formal communications (e.g. not in professional speeches, articles or books, but likely in conversation and similar communication)
Non-standard describes a usage (often a spelling) which is not the typically-accepted form but has some usage. It is typical for non-standard forms to show up more in informal communication, but this isn't exclusive.
