The promotion of an indirect object to direct object, illustrated by the sentence pair
Mary gave the book to John.
Mary gave John the book.
In a heavily case-marked language like Latin or German, the recipient John is always in the dative case, and the object passed across (the theme, as it's termed these days) is in the accusative case.

Traditionally in English the equivalent of the dative case is supposed to be a prepositional phrase beginning with 'to', and a verb is closer to its direct object ('the book') than to its oblique phrases ('to John', 'for his birthday', 'on Wednesday'). Hence the second sentence is regarded as shifted.

However, this analysis is not satisfactory. English hasn't got any distinction between accusative and dative in its morphology, and prepositions (appear to) govern the same case as verbs. What we have is a ditransitive* verb, one that takes two objects. The verb 'give' has two alternative constructions, one taking two objects directly, and the other taking one object directly and one after a preposition. It is tempting to say that 'give' has the option of taking two direct objects, and in 'Mary gave John the book', it is 'John' who is the direct object, or one of them.

As this conflicts with traditional grammatical terminology, we need to distinguish between semantic and syntactic roles. Regardless of how they are expressed, Mary is the AGENT, John is the RECIPIENT, and the book is the THEME: these are their semantic roles. Subject and object, however, are defined by their place in the syntax, and can be changed while preserving the meaning:

(1) Mary gave the book to John.
(2) The book was given to John (by Mary).
(3) Mary gave John the book.
(4) John was given the book (by Mary).
(5) */? The book was given John (by Mary).
In (1) there is a single direct object, and it can be passivized to (2), promoting the object to subject. In (3) there are two objects, the first of which can be passivized as in (4), but the second one can't be in (5) for many speakers.

There is still a syntactic distinction between the two roles, even when both are converted to 'direct' objects. The passive (4) is always acceptable, that in (5) only dialectally so. Also, you can't reverse the order:

* Mary gave the book John.
Not all verbs with two essential arguments can undergo dative shift. For example, 'put' must have two objects, and the THEME is direct, but the LOCATION role has to stay oblique (i.e. governed by a preposition):
Mary put the book on the shelf.
* Mary put the shelf the book.
* Mary put the book.
* Mary put on the shelf.
Even semantically very close verbs may behave differently. 'Donate' means 'give' and has the same roles, but can't dative-shift:
Mary donated the book to the museum.
* Mary donated the museum the book.
'To' is not the only preposition that can indicate the dative: 'for' has several roles, one of which is to mark the (intended) recipient.
Mary baked a cake for John.
(*) Mary baked John a cake.
If Mary baked the cake and gave it to John, or if she intended to but it was so good she ate it herself, you can dative-shift and say she baked him a cake. But if 'for' means on his behalf -- if John was supposed to bake it but overslept so Mary did it instead -- then you can't say the dative-shifted version for that meaning.

Not all uses of 'to' are dative: it can also indicate motion towards. It makes a difference.

Mary sent the parcel to Japan.
* Mary sent Japan the parcel.
Dative shift is ungrammatical here because the 'to' phrase isn't dative. Interestingly, you can sometimes read it as dative: in 'Mary sent Moscow the letter' it means she sent it some people in Moscow (perhaps her handlers), with personification of Moscow to be eligible as a recipient.

When we have two possible ways of expressing the same meaning, the choice of one or the other is typically for reasons of contrast or focus or emphasis. The subject of a sentence is usually the topic. In 'The book was given to John' you're talking about what happened to the book, and conveying new information that it was given to John. The two objects of a dative show some effects of new vs given:

(6) John gave the girl a book.
(7) John gave the book to a girl.
(8) ? John gave a girl the book.
(9) John gave a book to the girl.
Roughly, the definite article 'the' marks something known, and 'a' introduces new information. The known information comes first, so (6) and (7) sound normal, whether with or without dative shift. (8) sounds odd: you could say it, but it's not the most natural way, introducing new information 'a girl' then reverting to known information 'the book'. In fact it would sound better if 'the book' was something longer: 'John gave a girl the book he had been writing since May'. A heavy noun phrase can be moved to the end of a sentence so that you don't get distracted in it and lose track of the overall grammar of the sentence: this is called heavy shift.

(9) can be read with heavy shift -- 'John gave a book to the girl he was rather sweet on' -- but sounds fairly natural even without it. This slightly suggests this is the default construction in English, and that dative shift really is a shift of one actant into a higher position.

The expression 'did so' can substitute for a constituent, so the unacceptability of the first of the following shows that a dative-shifted object can't be a simple direct object; but personally I find the second odd too, rendering the shift analysis dubious.

* John gave the girl a book, and Mary did so a record.
? John gave a book to the girl, and Mary did so to the boy.

* The word 'ditransitive' seems to have a slight ambiguity: sometimes it means any verb taking two objects, sometimes only that 'double object' configuration in which the two objects are direct.

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