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B.Russell holds that certain classes of expressions are to be defined contextually, like definite descriptions or incomplete symbols in general, and some others are to be defined explicitly. What is the difference between the two types of definitions?

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It reflects the difference between two types of knowledge, knowledge by acquaintance and knowledge by description. "We shall say that we have acquaintance with anything of which we are directly aware, without the intermediary of any process of inference or any knowledge of truths". Russell's notion of acquaintance is extremely narrow, he reduces it to indexicals "I", "this", "here" and possibly "now". Only these can define explicitly.

Everything else is known only by description, which is some so-and-so, and to know it is to know that it is so-and-so and that so-and-so exists. For a definite description this so-and-so also has to be unique. Russell even counts proper names as implicit descriptions. Definitions of descriptions are contextual, Russell does not directly define their meaning, but rather defines the meaning of sentences containing them:"This is the principle of the theory of denoting I wish to advocate: that denoting phrases never have any meaning in themselves, but that every proposition in whose verbal expression they occur has a meaning". For example "the present king of France" is rephrased in sentences as "there is an x, x is now a king of France, and if y is also now a king of France then y is x". Although "the present king of France" sounds like an object, in the rephrasing it is essentially dissolved into predicates, which is why Russell sometimes calls descriptions incomplete symbols.

See Knowledge by Acquaintance vs. Description and Russell On Denoting.

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From the "formal" point of view, the basic difference is the following (ref to : Alfred North Whitehead & Bertrand Russell, Principia Mathematica to *56 (2nd ed - 1927) :

For an explicit definition, like the abbreviation [page 11] :

p ⊃ q . = . ¬p v q Df.

we can always replace the definiendum (the newly introduced symbol; in this case : ) with its definies.

In the case of a contextual definition, like the incomplete symbol [see page 67] :

(ix)(φx)

we cannot "remove" them, but only the "contexts" where they occur :

all phrases (other than propositions) containing the word the (in the singular) are incomplete symbols: they have a meaning in use, but not in isolation. [...] It follows from the above that we must not attempt to define "(ix)(φx)" but must define the uses of this symbol, i.e. the propositions in whose symbolic expression it occurs.

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