אשר

Abarim Publications' online Biblical Hebrew Dictionary

אשר ישר

The form אשר ('sr) occurs in two different ways: There's the verbal root אשר ('ashar), which indicates progression, and there's the particle אשר ('asher) that indicates relation. Whether the two are etymologically related isn't clear, although there seems to be an obvious intuitive connection. And then there is the verb ישר (yashar), which appears to be etymologically related and certainly is so in meaning. Ultimately, these words may even have to do with the base root שרר (sharar), which speaks of rigidity resulting from the absorption and retention of liquids or liquidity (anything from wealth to knowledge).

The verb אשר ('ashar) generally indicates a decisive progression (Proverbs 4:14, 9:6) or a setting right (Isaiah 1:17). On occasion it's used in the negative (literally: Isaiah 3:12; leading someone "straight astray"), but most often it's positive. So positive even that this verb's secondary meaning is that of being or being made happy (Psalm 41:2, Proverbs 3:18), or even being deemed or called happy (Genesis 30:13, Job 29:11, Psalm 72:16).

The derivatives of this verb are:

The masculine nouns אשר ('esher) and אשר ('ashar), meaning happiness or blessedness (1 Kings 10:8, Psalm 32:1, Isaiah 30:18). This word most often occurs in the plural construct (that's אשרי or 'happinesses of ...' or 'happinesses to ...', meaning 'happy is ...'), which is not all that odd. Hebrew uses plural to express emphasis, and so, on occasion, does English: 'very, very good times'.
The masculine noun אשר ('osher), meaning happiness as well, and only used in Genesis 30:13, in the construct באשרי (b'asheray), meaning in my happiness.
The feminine noun אשור (ashur), meaning a step or a walk; a going (Job 23:11, Psalm 17:4).
The feminine noun אשר (ashur), also meaning a step or going (Job 31:7, Psalm 17:5, 17:11 only).
The feminine noun תאשור (te'ashur), denoting a kind of tree, namely the box-tree, which appears to be distinguished by the upward direction of its branches; a happy-tree, or perhaps a straight-up tree (Ezekiel 27:6 only).

אשר

The relative particle אשר (asher), generally meaning who or which, looks like it came straight from the above root, but apparently, that's not so. None of the sources even hints at it, although BDB Theological Dictionary declares its "origin dubious". Our particle occurs in Moabitic with identical meanings but (as HAW Theological Wordbook of the Old Testament notes) it has been found only once in the vast collection of Ugaritic texts that has been unearthed. Since Hebrew and Ugaritic are closely related, this absence of our particle in Ugaritic seems to disarm BDB's objection against one of two plausible theories of its origin:

This one theory suggests that our word אשר (asher) originated in a word that in Arabic means footstep or mark (which brings it very close to the previous root indeed), then went on to serve as a marker of locality (a place), then acquired the meaning of there and where, and evolved on to become the relative mark we know it as. BDB states that "the chief objection to this explanation is that it would isolate Hebrew from the other Semitic languages, in which pronouns are formed regularly from demonstrative roots".
The other theory BDB lists involves an unlikely exchange of the letter ל (lamed) of an assumed construction אשל ('sl) for the ר (resh) of our particle אשר (asher). BDB admits that, despite the objection, the previous theory remains most plausible.

The particle אשר (asher) occurs all over the Old Testament (instead of simply submitting a number, HAW Theological Wordbook of the Old Testament excitedly reports that Mandelkern's concordance lists "twenty pages, small print, four columns to each page" of occurrences of אשר, asher).

Our word primarily expresses relation: this which that, or he who such and such. In some cases it may express result: so that if a man could number the dust . . . (Genesis 13:16), or purpose: in order to find favor (Ruth 2:2), or causality: because of their sister (Genesis 34:27), or concession: although you made me see trouble (Psalm 71:20).

Our word very often comes with its own preposition, creating even more nuance and meaning:

With ב (be), meaning in it forms the word באשר (b'sr), which means in which, or in that (Genesis 39:9, Isaiah 56:4).
With מ (me), meaning from, it forms מאשר (m'sr), which means from that which (Genesis 39:1, Joshua 10:11).
With the comparative particle כ (ke), meaning like, it forms כאשר (k'sr), which means according as, or simply as (Genesis 34:12, Exodus 10:10, Isaiah 9:2), or it means in so far as or since (Genesis 26:29, Numbers 27:14), or when (Genesis 18:33, 1 Samuel 6:6).

Closely synonymous to the relative particle אשר ('asher) is the relative prefix ש (shi). Scholars appear to have concluded that this particle and prefix share no etymological root, but the argumentation surrounding this conclusion is sketchy at best. Whether coincidently or not, the particle אשר ('asher) and prefix ש (shi) are as alike as the particle על ('al) and the prefix ל (le), and the particle כי (ki) and the prefix כ (ke).

ישר

The verb ישר (yashar), generally means to be level or straight. It's used in four distinct ways:

Literally, of a road being straight (1 Samuel 6:12), or smooth (Isaiah 40:3).
Ethically; of a just or virtuous life style; blameless (Proverbs 11:5), or discerning (Psalm 119:128).
To be right in the eyes of someone, which means to obtain this person's approval (Judges 14:3).
Tranquility or harmony: of a soul being at peace (Habakkuk 2:4)

The derivatives of this verb are:

The adjective ישר (yashar), meaning right or upright (Isaiah 26:7, Exodus 15:26).
The masculine noun ישר (yosher), meaning uprightness (Proverbs 2:13, Job 6:25).
The feminine noun ישרה (yeshara), also meaning uprightness (1 Kings 3:6 only).
The noun מישר (meshar), means uprightness, straightness, mostly in an ethical sense (Isaiah 26:7, Proverbs 8:6).
The noun מישור (mishor) means a level place or uprightness mostly in a geographical sense (1 Kings 20:23, Psalm 26:12).

An obvious demonstration of the kinship of these two verbs can be found in the two names Asharelah and Jesharelah, which are applied to the same person.