The 'court', in a British social context at least, refers to the social establishment of the monarch. The terminology is still used officially today for ambassadors to the United Kingdom, who remain ambassadors to the 'Court of St James'; this refers to the palace, at one time the main seat of the monarch, and reflects the fact that the King or Queen's court was the centre of political life in the country. This has practically eroded almost entirely with the advent of constitutional monarchy, but as Head of State ambassadors are still nominally sent to the court.

In social parlance, the 'court' refers to the monarch's establishment as well, albeit more varied and less official in application. Different strata of society, from gentry to aristocracy, would be presented at court for a number of purposes, from the medieval period up to the present day, in a way. For instance, Sir William Lucas in Jane Austen's Pride and Prejudice (set in the late eighteen century) has been presented at court to receive a knighthood in connection with his civic rôle. This means he has met the King at a function, presumably an investiture ceremony. It is notable enough a distinction that he comically mentions it to anyone and everyone, and seems to represent the apex of his social credentials.

Young ladies, by contrast, were sometimes presented to the King by socially established parents when 'coming out' (i.e. entering society when of marriageable age), or with their husbands on their marriage. This practice denoted one's place in the upper echelons of the hierarchy; access to the monarch on a social level remains even to this day a much coveted social symbol in certain sections of society. For literary reference, see Nancy Mitford 'Love in a Cold Climate', where Lady Montdore frequently discusses considerations for presentation 'at court', delightfully dreadful snob that she is.

In short, 'the court' means a function at which the monarch is acting in the capacity of host or hostess. 'Presentation at court' means you've been introduced to the monarch, and you have duly exchanged pleasantries. Whether or not the monarch subsequently would have any recollection of this is doubtful, and differs from case to case; but you would almost certainly remember yourself, and ensure those in your own society remembered it too!

EDIT: Re-reading this answer, I am concerned it suggests women were not 'presented' in their own right, only as daughters or wives. Obviously many women have been presented for their own accomplishments, more now than ever; but, sadly, this is a somewhat later phenomenon for the most part, and your reference to period dramas directed my thoughts to the more historic contexts, where women were unfortunately often compelled to use their connections to ascend the social hierarchy, a process in which 'presentation' usually formed a part.