The first ten days of Tishrei, from the first of Rosh Hashanah until Yom Kippur inclusive, are known as the Asseret Yemei HaTeshuvah - the 'Ten Days of Repentance'. These are days of mercy and forgiveness, days dedicated to repentance, to examining one's conscience and, specifically, to amending one's ways.
'Although repentance and prayer are always effective, they are even more effective during the Asseret Yemei HaTeshuvah when they are accepted instantly'. (Maimonides)
'Isaiah wrote, "seek G-d when He is available. Call on Him when He is near". When is G-d available? When Israel repents. And when is He near? When He is called on sincerely.' (Midrash Tanhuma)
The seven intermediate days between Rosh Hashanah and Yom Kippur are termed 'Bein Kesseh le'Assor' (Between Kesseh and the Tenth), Kesseh being a name attributed to Rosh Hashanah (Psalm 8:14), and the tenth being the 10th of Tishrei (Yom Kippur).
These two names (for the ten days and the seven intermediate days) are a reminder about the importance of these days are, and how a person is suspended between the two days of judgement: Rosh Hashanah, when the individual is inscribed ('written') and Yom Kippur when the decree is confirmed ('sealed').
The Sabbath between Rosh Hashanah and Yom Kippur (there is only ever one of these) is called Shabbat Shuvah (the Sabbath of Return), because the Haftarah is read from Hosea 14:2 which begins with the word Shuvah ('Return!'). It is often known as the Sabbath of Repentance, because it falls in this period and to 'return' means 'repent'.
There are ten days to set right wrongs committed by the House of Israel and there are several significant and related references to the number ten:
They have not fulfilled the Ten Commandments;
Harm has been done to the world that was created by means of ten instructions (the word 'said' occurs ten times in the first chapter of Genesis);
They have not fulfilled the obligation placed on them by the ten miracles that were preformed for them in Egypt ;
The ten plagues inflicted on the Egyptians.
'Return, Israel, up to G-d your G-d, .... and return to G-d.'
First we return up to G-d, and then we are able to return to G-d.