The acronym Elul in the Hebrew letters stands for ‘I for my uncle and my uncle for me”.
The last month of the year tunes the hearts for the preparations towards the new year. Some prepare for the “Days of Awe” and some call them “Tishrei Holidays”. Either way, Elul serves as a period of transition between the passing year and the incoming year.
The Sephardic Jews rise early in the morning to say their Selichot prayers from the beginning of Elul, and all ethnic communities start blowing the shofar as of that day until Rosh Hashana Eve.

The biblical text does not link Elul to Rosh Hashana, and does not even link Rosh Hashana to Yom Kippur.

The biblical text does not connect Elul with Rosh Hashana, nor between Rosh Hashana and Yom Kippur. But with the help of the Midrash one can connect in a single - story sequence Shavuot - the fast of Tammuz - Rosh Chodesh Elul - Yom Kippur. From the details of the plot drawn from various parts of the Bible, it transpires that Moses ascended to Mount Sinai not once, not only twice, but three times, and in each of them he stayed there for 40 days.

 

And they made me a temple and I resided amongst them.
When was Moses told this episode of Mishkan?
On the Day of Atonement itself, even though the Mishkan's passage precedes the deed of the golden calf.

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Rabi Yehuda Barabi Shalom said:
In the Torah, there is no later or earlier time, as said: There is no early and late time in the Torah, as it is written: "Move from the wagons, you shall not know" (Proverbs 5: 6), thrown are the paths of the Torah and its stories. Oh, on the Day of Atonement Moses was told: make me a temple.

From where?
Since Moshe went up on the sixth day of the Month Sivan, and stayed there forty days and nights, and then another forty and another forty, hence a hundred and twenty.
And on the Day of Atonement the Holy One, blessed be He, said to him, "Make me a sanctuary and I shall dwell among them, so that all the nations may know that they have been atoned for the act of the golden calf." This is called the Mishkan of Testimony, which is a testimony to all the people of the world, that the Holy One, blessed be He, dwells in your Temple.

On the sixth day of Sivan Moses went up to receive the Torah, and the people heard the ten Commandments. 
On the 17th day of Tamuz, 40 days later, he went down from the mountain, saw the Golden Calf and broke the boards.
On Tamuz he went up Mount Sini again to ask for mercy. He stayed there another 40 days and went back to the people.
On the first day of Elul, he went up mount Sini once again to receive the second set of boards.

On the 10th day Tishrei, 40 days since Rosh CHodesh Elul, Moshe came back with the second set of boards. This is Yom Kippur, and it is the day when God forgave the people for the calf sin: “And God said I have forgiven as you said”.

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