Abarim Publications' online Biblical Hebrew Dictionary
יום
The important masculine noun יום (yom) means day, and comes with one derivation, namely the substantive and adverb יומם (yomam), meaning daytime (Jeremiah 15:9, Ezekiel 30:16) or by daytime (Numbers 10:34, Isaiah 4:6). The word יום (yom) in plural is spelled ימים, and the pseudo-genitive plural form meaning 'days of' is spelled ימי. Also note the similarity with the noun ים (yam), meaning sea; the plurals of both these words are spelled ימים.
Linguistically, the noun יום (yom) meaning day is used in much the same way as we use the word day in English, but with a few major differences. As a concept, however, the Hebrew day is something wholly different than a modern day.
The Biblical concept of the day
In Hebrew, nouns derive from verbs but it's not known from which verb the noun יום (yom) derives or which motion it expresses. But it does express some kind of motion or action, and not a section of time that starts at 00:00 and lasts until 24:00 (see for a discussion of this our article on the Yom Problem).
Our noun is mostly used to indicate the light-part of a solar day, or rather, the part during which one can work, travel and trade. It's opposed to לילה (layela) and ליל (layil), meaning night (Genesis 7:4, 1 Kings 8:29, Amos 5:8). Although either rare or non-existent, the concept of day may also be used to describe an indefinite period of light (peace, wisdom, etcetera) following an indefinite period of darkness (war, ignorance, etcetera; see the concept of this manifested in Isaiah 9:2, although our noun יום is not used there). In a daily sense there is no particular relation between a day and either the night ahead or the night past but in theory, night came first and light came after (Genesis 1:2-3). A day like the Sabbath (which is the only day of the week with a name in Hebrew), starts at dusk of the previous day but that's because no work was done in the night anyway. Then it goes on all day, and theoretically ends at dusk of that day (see Leviticus 23:32), but in the night following, again no work is done, so for all practical purposes, Sabbath lasts from dusk until the second dawn.
Our noun is also used as a unit of time, but that mostly in regard to the amount of work that can be done; labor-time rather than clock-time. Hence a one-day journey does not primarily refer to the amount of hours required to get somewhere, but rather to the distance that can be covered without having to stop travelling, bed down and start up again the following morning (Numbers 11:31, 1 Kings 19:4). The same goes for a three day journey, which is the distance that can be traversed when allowing for two nights; anything between a little over one 'day-time' up to three 'day-times' travel (Genesis 30:36, Exodus 3:18). This perhaps anti-intuitive use of time also explains why Jesus was said to have been in the grave three days, while in fact he was buried on Friday night and rose on Sunday morning. The same goes for a seven day journey, during which the traveler has to account for at least one Sabbath day (Genesis 31:23, 2 Kings 3:9).
Similarly, the plural form, ימים, literally meaning days, does not so much denote a period on the calendar but rather work done or things achieved in whatever time frame: a short while (Genesis 27:44, 1 Kings 17:15), one human life (Genesis 6:10, Joshua 24:31, 1 Samuel 25:28), or one working career (Genesis 14:1, Judges 5:6, 2 Kings 20:19). As also demonstrated by the use of the name Sabbath both for a day and a year (Leviticus 25:5), the plural ימים may also be used to mean year (Exodus 13:10, Judges 17:10). The regular word for year is שנה (shana).
The ancients discussed time mostly as the cyclical phenomenon that comes in days, months and years, but they also understood that time had a beginning and an end, and we rarely pause to reflect what an incredible insight this was, because the fact that the universe has a beginning can't be readily observed. For reasons that are not too well understood, they divided all time up into seven periods which they also called days (Genesis 1). This is generally referred to as the creation week (which hasn't ended yet) but why the ancients figured that creation is permeated with the number seven is unclear. Days, months and years can be observed, but where the week comes from is a mystery (but read our introduction to Scripture Theory for some hints). The invention of the Sabbath, the weekly day-off for all humans and domesticated animals, is by no means less mysterious because nothing in nature has time off.
Besides the Sabbath, there are at least two more days with a name, namely Yom Kippur (see the link below) and the Day of YHWH (Isaiah 2:12, Amos 5:18, Zephaniah 1:7).
Although it's not specifically mentioned, the latter probably denotes the seventh day of creation. This is perhaps difficult to match with the chronology culminated in Genesis 1:31, but the idea is that when the Lord stopped creating and entered his Sabbath, he had created everything and the potential of whatever creation could evolve into. That would mean that we are still living on the sixth creation day, forming into whatever we can be. When we are fully formed, according to our potential in this world, we too enter God's Sabbath (or that's the plan; see Psalm 95:11, Hebrews 3:11). The Day of YHWH isn't so much coming to us as we are going to it — perhaps a bit of a confusing expression, but maybe on a par with saying that a city "whooshes by" when in fact we are in a whooshing train looking out the window at a stationary city.