In some cases אֶחָד is used to describe unity, both bringing together and the state of oneness.
The word is a defining characteristic in the Shema:
שְׁמַ֖ע יִשְׂרָאֵ֑ל יְהוָ֥ה אֱלֹהֵ֖ינוּ יְהוָ֥ה אֶחָֽד
Hear, O Israel: The LORD our God, the LORD is one. (Deuteronomy 6:6 ESV)
Here it unites YHVH and Elohim: יְהוָ֥ה אֱלֹהֵ֖ינוּ יְהוָ֥ה. "One" could be The Name, or Elohim individually or in combination as YHVH-Elohim or Elohim-YHVH. Since all identify the True God, they are also collectively, One.
While there is nothing in Deuteronomy or other liturgical passages to indicate the Shema was to be given a place of central theological importance, in the late Second Temple period it was cited as the greatest command1. The New Testament recording of the Shema also reflects oneness:
And one of the scribes came, and having heard them reasoning together, and perceiving that he had answered them well, asked him, Which is the first commandment of all? Jesus answered, “The most important is, ‘Hear, O Israel: The Lord our God, the Lord is one. (Mark 12:28-29 ESV)
The New Testament citation of the Shema unites Lord and God in a similar manner: Κύριος ὁ Θεὸς ἡμῶν Κύριος εἷς ἐστιν.
I believe the best example of unity is found in the first use of אֶחָד. Genesis 1:5 uses the word in the context of "day" and "night" allowing the natural world to serve as an example:
God called the light Day, and the darkness he called Night. And there was evening and there was morning, the first day י֥וֹם אֶחָֽד. (ESV)
God called the light day, and the darkness He called night. And there was evening and there was morning, one day י֥וֹם אֶחָֽד. (NASB)
אֶחָד is both the first day and one day. It is the day which marked the first evidence of God’s work of creation. At the same time it is one single day in God’s process of creation.
One aspect of the nature of unity is conveyed in the context of י֥וֹם אֶחָֽד. Light was called “Day” and darkness was called “Night.” י֥וֹם אֶחָֽד consists of a period of light and dark. In other words, אֶחָֽד unites the darkness and light into a single period of time. As noted here, Is there any significance to the shift between cardinal and ordinal numbers in Genesis 1? you would expect to see רִאשֹׁ֔ון instead of אֶחָֽד in verse 5. But while רִאשֹׁ֔ון would correctly describe the first day, it leaves open the possibility that only the period of light is considered. Thus one aspect of unity of אֶחָֽד is that it brings together two things which have significance independent of the other.
Another use of אֶחָֽד gives an similar example:
Therefore a man shall leave his father and his mother and hold fast to his wife, and they shall become one (אֶחָֽד) flesh. (Genesis 2:24 ESV)
Just as in the first day, אֶחָֽד describes a unity which brings together two equally independent and capable of standing alone and does so in a way that also describes something unique.
In Genesis 1:5, אֶחָֽד adds continuity to the unity:
God called the light day, and the darkness He called night. And there was evening and there was morning, one day י֥וֹם אֶחָֽד. (NASB)
One day י֥וֹם אֶחָֽד is not recognized until after there is a second period of darkness and a new period of light begins. As a day contains both dark and light, it is complete when the next period of darkness starts. Just as one would expect to have the first day described as רִאשֹׁ֔ון, one would expect the first day to end and the second day to start with the onset of the next darkness. However, י֥וֹם אֶחָֽד is not described until both the darkness and the light of the next day are affirmed:
Describing י֥וֹם אֶחָֽד after there is morning recognizes the reality the cycle is repeating. This conveys continuity. That is, one day is not only a period of darkness and light; it also connects the next periods of both dark and light; the single phrase describes a united period of dark and light in a way that includes continuity with the next periods of dark and light.
So in the case of Genesis 1:5, אֶחָֽד not only brings together what could be considered as different to make something unique in its own right, it does so in a way that conveys continuity both with the different elements (darkness and light) and how they were brought together (a day). It is used to describe a continuous unity.
1. Bernard M. Levinson, The Jewish Study Bible (2004) p.379