Good question. There are basically two main “types” of angels according to Judaism. One type is an angel made by God to act as a messenger or agent in this world. The other type is an angel that is formed when a person does a mitzvah.
The Hebrew word for "angel" is "malach." The word "malach" is related to the word "melacha," which means "task." Hence, a malach is an agent or vehicle which accomplishes a task. The English word "angel" comes from the Greek word "angelos" meaning "messenger" or "agent."
Therefore, a "malach" can be a wind, a person, or a purely spiritual force. The angels such as those who spoke to Abraham and Jacob were purely spiritual forces which appeared in human form.
Regarding the second type, our Sages state that when you perform a commandment you create an angel that accompanies you. Maimonides explains that this refers to the spiritual and intellectual concepts that influence the person as a result of performing the commandments.
According to Maimonides, the degree of Divine Providence you experience is directly proportional to your attachment to God, and therefore the commandments which create that attachment are the true "guardian angels" of a person.
Sources:
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Tehillim 104, Commentary by Rabbi Avraham Chaim Feuer
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Zohar, Raiya Mehemna Parashat Shelach & Tractate Avodah Zarah 17a
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Maimonides, Mishneh Torah, Laws of Mezuzah, 6:13
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Maimonides, Guide for the Perplexed, Section 3
