
When someone is at your beck and call, they're available to help you any time, like a servant.
Outside of this expression, the word beck is rarely used in English anymore. It's originally a thirteenth-century abbreviation of beckon, and on its own came to mean the sort of nod or gesture you make with your head to bring someone closer to you. And a call is a loud utterance to get someone's attention.
These examples show how beck and call is generally used:
"Oh, I'm sorry I can't be at your beck and call all the time." (The Marrow Thieves)
But there was also a docility about his students – answering to his beck and call and unwilling to challenge him. (The Guardian)
Beckon call is incorrect because it does not make sense grammatically. Both beck and call are nouns, but beckon is a verb. If you beckon call, you would be...summoning a call? In the examples above, notice how beck and call always serves as a noun phrase. Being at someone's beck and call is the same as being at their service. The following sentences show some misuses of beckon call. Notice how the verb beckon doesn't make sense in the examples below; it sounds almost as though it's being used as an adjective to modify call — but beckon can't be used as an adjective.
Lying in her bed, she could have the doctors and nurses at her beckon call, either by pressing a button or feigning an outrageous symptom that warranted further attention. (Yale Journal for Humanities in Medicine)
In fact the very day before the attacks, on September 10th, Attorney General John Ashcroft, at the beckon call of Bush, denied the request for money. (Susquehanna University Crusader)
It's easy to understand why people make this error: beck and beckon are closely related, and beck and sounds a lot like beckon when spoken aloud. Just remember that there's an and in the middle of this expression and that the and joins two nouns. Used correctly, this phrase is an excellent way of describing a relationship where one person is available on a moment's notice to fulfill another's every desire or need.