Is "yesterday" a noun, an adjective or an adverb?

Solution 1:

They can work as nouns or adverbs.

For example:

• "Yesterday was a great day"; here, yesterday works as a noun.
• "I will do that tomorrow"; here, tomorrow works as an adverb.

Solution 2:

Get ready for more mixed signals. The Cambridge Grammar of the English Language (CGEL) gives an analysis that differs dramatically from the other answers here.

It says that yesterday, today, tonight, and tomorrow are pronouns. The evidence:

• Like I and you, they're deictic. Which day yesterday is depends on the context of the speech act, i.e. when you say it.
• Unlike common nouns, they don’t take determiners. You can’t say The yesterday was great.
• Unlike adverbs and prepositions, they have a possessive form. Compare: [Usually’s / After’s / Now’s / Yesterday’s] performance was great.

(It doesn't mention Shakespeare’s “...And all our yesterdays have lighted fools the way to dusty death.” which shows yesterday acting a lot like a noun. I suppose they’d say that’s just Shakespeare playing with words.)

In a case like "I have an important meeting tomorrow," it seems as if they're nouns.

CGEL spends several pages on “temporal location expressions”. They are sometimes but not always adverbs. Several examples are given of noun phrases that specify time: I have an important meeting [Tuesday / tomorrow / the day after tomorrow / every day / next month / right this minute]. That is, certain noun phrases can be tacked onto a sentence in just the same way as an adverb or a prepositional phrase.

But what about "Yesterday afternoon?"

Here the pronoun yesterday functions as a determiner. This is not something pronouns normally do; it's an oddball case.

Determiners include the bolded expressions in twelve angry men, my red tennis shoes, a sandwich, your father's truck, three or four billion dollars. A singular count noun generally needs a determiner in front of it if it's going to function as, say, the subject of a sentence. Compare: [This afternoon / Yesterday afternoon / Afternoon] was great.

Apparently the days of the week can also serve as determiners this way: Sunday afternoon.

Solution 3:

They are actually both considered adverbs in the uses from your example.

One bit of evidence for this is that you could replace tomorrow in your example sentence with other time adverbs, or a word like frequently or daily, but you couldn't replace it with something that is a noun only, like "office", or even "5 PM".

Solution 4:

Words don't in general belong to a single part of speech (POS), that is POS is not in general a lexical property of a word, but they are normally associated with a particular dominant role/POS which in this (*day) case is Noun (or NP), although like any (prepositional) noun phrase, particularly one locating in space or time it can act as an adverb. Nouns, parts of the body, time points or durations, etc. can even act as verbs (when there is no more obvious and appropriate verb to use - he shouldered him aside and headed the ball into the goal; he minuted the discussion). The true English adverbs are a closed class or marked with a functional morpheme (usually -ly) - he sidled slowly, crablike, into the room.

Thus here... Yesterday, today and tomorrow are nouns that can act as complete noun phrases as they are reductions of forms that include a determiner. See e.g.

• http://www.etymonline.com/index.php?allowed_in_frame=0&search=yesterday
• http://www.etymonline.com/index.php?allowed_in_frame=0&search=today
• http://www.etymonline.com/index.php?allowed_in_frame=0&search=tomorrow

In general, (prepositional) noun phrases can act adjectivally and adverbially, and in particular when the noun is already marked for place or time, the preposition and/or article can often be omitted as definiteness is understood, particularly in sequences of places and/or events. In journalism and diaries, place, time and byline conventionally omit the propositions. It is not so much that there is anything special about *day (except you normally won't add another determiner to today/tomorrow/yesterday as complete deictic noun phrases centred on the time of speaking/writing). When you do add a determiner it recentres the deixis (which is an attribute describing a word that locates something in the space-time context).

• Sunday, I visited some museums.
• London, I was exhausted after the flight, but Edinburgh, I visited every museum.

• London Sunday, another explosion rocked the underground as evening commuters...

• all my tomorrows will be spent with you!

• every tomorrow is a new day!
• the tomorrow I'm looking for ...

• my tomorrow isn't looking too promising!

• all my yesterdays are as nothing now I have met you!

• yesterday evening's train ...
• last night's train ...
• yesterday's events ...
• the week's events ...
• this week's evening events ...

In speech when a multiword noun phrase is used as an adjective rather than a single adjective, the extra words are conventionally hyphenated prefixes in written form to show that the additional words are not modifiers of the main noun, but of the adjectival noun (unless there is no difference in semantics):

• early Iron-Age artefacts
• an in-your-face kind of guy
• an on-the-ball comment
• a last minute decision

The last example is sufficiently frozen that it is still common without the hyphen although it does now occur with the hyphen. The more extreme version of this wordifying push is when the hyphen/space gets dropped completely as has happened with today, tomorrow and yesterday. 30 years ago it was most often 'to-day', and 50 years ago even 'to day' and 100 years or more ago 'the day' or 'this day' (sufficient unto the day is the evil thereof; give us this day our daily bread) - this has changed twice during my lifetime!