In Sh'mos 4:10, after Hashem instructs him to speak to the Jews, Moshe describes himself to Hashem as not an "אִישׁ דְּבָרִים", literally "man of words", and as "כְבַד פֶּה", literally "heavy of mouth", and "כְבַד לָשׁוֹן", literally "heavy of tongue". In 6:12, after Hashem instructs him to speak to Par'o, Moshe describes himself to Hashem as "עֲרַל שְׂפָתָיִם", close-to-literally "uncircumcised of lips" (Rashi: "sealed-up of lips").
What does each of these four refer to?
Tol'dos Yotzchak (by Rabbi Yitzchak Karo, uncle of the Bes Yosef), in his commentary to 4:10, says that "כבד לשון" refers to an impediment in pronouncing the so-called tongue letters, דטלנת; "ערל שפתים", the lip letters, בומף; and "כבד פה", the rest. ["לא איש דברים", then, would seem to be an all-embracing expression.]
According to Ibn Kaspi, "לא איש דברים" means that Moshe was not an eloquent speaker, he was literally not a man of words. This was relevant because God was asking him to be a leader, and good public speaking skills are often thought of as crucial to such a role. No one would ever get elected president or prime minster if they couldn't deliver a good speech.
"כבד פה" and "כבד לשון" on the other hand, are actual speech impediments. (According to the R' Chananel quoted in the footnote in the link, "כבד פה" would refer to the letters זרש"ץ and "כבד לשון" to the letters דטלנ"ת.)
Now, "ערל שפתים" could refer to a third speech impediment (which would be the "lip letters" בומ"ף) like the opinion of R' Yitzchak Karo quoted in @msh210's answer and what seems to be the opinion of Abarbanel, but since Moshe didn't mention it originally when talking to Hashem like he mentioned the others, I prefer the Ralbag's understanding, that it means simply that Moshe is not a good speaker in that he often has trouble expressing himself through speech, often has trouble finding the right words to express the idea he is trying to convey. In this respect, "ערל שפתים" is kind of an antonym of "איש דברים".
On a clinical level, it almost sounds like apraxia of speech.
http://www.apraxia-kids.org/site/apps/nlnet/content.aspx?c=chKMI0PIIsE&b=839037&ct=837215
This is my take on Moshe's speech "impediment" based on "hekesh" (comparison of verses): In Parshat Shmoth (4.10) Moshe describes himself to Hashem as, literally "not a man of words." The word "d'varim" is related to prophetic speech, concerning the Commandments of HaShem, not ordinary (amar) speech which explains the prophetic words of HaShem. The former, devar is precise, direct "guarded" speech; whereas the latter "amar" are the details of the "direct" speech.
Here, Moshe's speech was limited to prophetic words and deeds.
For instance, in 6:12 we read that Moshe said to HaShem, "Behold, the children of Israel did not hearken to me. How then will Pharaoh hearken to me, seeing that I am of closed lips?" Here, Moshe describes himself to Hashem as "עֲרַל שְׂפָתָיִם", close-to-literally "uncircumcised of lips" (Rashi: "sealed-up of lips").
In fact, strength for the argument that debeer / devar is specifically related to the mitzvoth comes from the passuk, (6.13)
"So the Lord spoke ("devar") to Moses and to Aaron, and He commanded them concerning the children of Israel and concerning Pharaoh, the king of Egypt, to let the children of Israel out of the land of Egypt."
Because Moshe needed an interpreter (his brother Ahron) we can easily derive that he had a barrier to being understood. (7.1):
"The Lord said to Moses, "See! I have made you a lord over Pharaoh, and Aaron, your brother, will be your speaker (נְבִיאֶֽךָ "prophet")."
If his speech was limited to prophetic speech, this would explain the Targum.
Indeed, the first-century C.E. Aramaic translation of the Bible known as Targum Onkelos, which preserves some of the oldest rabbinic interpretations to have come down to us, characterizes Moses’ speech as profound, rendering k’vad peh as yakir mamlal, “weighty of speech,” and k’vad lashon as amik lishan, “deep of tongue,” turning Moses’ negative self-description into a positive one.
Thus, we might read Moshe's protest as, "I am not a man of [prophetic] words."
In another instance, if Moshe were speaking specifically to the Children of Yisrael, the passuk would read, "Va'Yomer HaShem eel Moshe Laymor, dabayr.... B'midbar 15.37
The Midrash (Deut. Rabba 42, Tanchuma Tzav) says the difference between the two is mainly a difference of tone. It considers DABER a harsh way of speaking and AMAR as being softer. Others say that DABER is used for elaborate or detailed explanations which AMAR is more direct and to the point. Rabbi Elijah of Vilna (Aderet Eliyahu) suggests that DABER is used for commandments that are spelled out in the written Torah whereas AMAR is used for those commandments found in the oral Torah. There is a sense that DABER suggests that there is a distance between the speakers and AMAR communicates closeness and intimacy. Perhaps God feels closer to his priests then to the man on the street.
The Ten Commandments are called עשרת הדברות – [Aseret HaDibrot] – the 10 utterances, which derives from the word DABER. They are short, concise and to the point.
In finding other support for my position that daber / devar is related to prophetic speech and the mitzvoth, I offer Jeremiah 1.4, 6 and 7 (And the word [daber] of the LORD came unto me, saying;" "Then said I: 'Ah, Lord GOD! behold, I cannot speak [daber]; for I am a child.'" "But the LORD said unto me: say not: I am a child; for to whomsoever I shall send thee thou shalt go, and whatsoever I shall command thee thou shalt speak [te'daber]." )
The question of the four "impediments" posted above by MSH410 are really only one speech impediment.
The Midrash relates that, when Moshe was a child in Pharoah's court, he continued to reach for the crown and was put to a test. When he reached for the crown during the test, an Angel directed his hand to grasp a coal and he put the coal in his mouth scorching his tounge. This saved his life but resulted in his speech impediment [which Rashi relates as stuttering].
