Why do Jews bow in prayer? Bowing is an expression of humility and submission to G-d (Mi Yodea). Note that it is forbidded to pray to something other than G-d, be it an idol, a thing or a person. Jews only bow to G-d.
There are a few places that you bow during prayers. I will list them here in the order they should appear in a service. At each of the points you should be standing. It doesn’t matter whether you pray with a group or alone, aloud or silent, you should bow in every case. There are two types of bows, a simple one from the waist (a "normal" bow), and a more special one where you first bend your knees slightly and then bow from the waist (animation from Judaism 101: Jewish Liturgy):
The first bow is right at the start of the service, during the community’s answer to Barchu, the call to prayer by the prayer leader. This is a normal bow from the waist. You need only bow for as long as it takes you to say the phrase "baruch ad-nay" ("blessed are you, G-d"). You need not wait until the prayer leader has finished repeating the repetition.
There are four places during the Amidah where you should bow. The first two are in the first brachah (blessing), the other two during the second-to-last brachah. Why exactly these two? Well, the first because it is the beginning and we are closest to G-d then, the other one because it is where we are thanking G-d for our existence which is the basis for everything else (Mi Yodea).
The first brachah of the Amidah is called "Avot" ("patriarchs"). It starts with the words "baruch ata ad-nay el-heinu veEl-hei avoteinu" ("blessed are you, L-rd, our G-d and G-d of our fathers"). You bow during the first two words and stand again at the third. The last sentence of the brachah has "baruch ata ad-nay magen avraham" ("blessed are you, L-rd, shield of Abraham"), same procedure applies there. This is the knee-waist version of bowing in both cases.
The second-to-last brachah is called "Hoda’ah" ("Thanksgiving"). It starts with "Modim anachnu lach sheAtah hu ad-nay" ("We thank you, for it is you who is the L-rd"). You start the bow at the first word and only stand again at "ad-nay". As I have learned it this is a simple bow, but some other sources have it as a knee-waist bow, so this might be dependent on your community’s tradition. The brachah ends with "baruch ata ad-nay haTov shimcha veLecha na’eh lehodot" ("blessed are you, L-rd, the good one, it is fitting to thank you"). Same procedure as in the first brachah, do a knee-waist bow during "baruch ata" and then stand. When the Amidah is repeated, some people stand up and bow for this brachah, other don’t.
After the Amidah, there is a short silent meditation. It contains the words "oseh shalom bimromav, hu ya’aseh shalom aleinu, veAl kol Israel, veImru amen" ("he who makes peace on the heights, may he make peace upon us, and upon all Israel, and say Amen"). Do a simple bow to the left for "oseh shalom bimromav", to the right for "hu ya’aseh shalom aleinu" and to the front for "veAl kol Israel, veImru amen". This is how we take leave from our king now that the prayer is finished. [As an aside, it is customary to take three steps forward at the beginning of the Amidah, just before this sentence and the bowing would be the time to take three steps backwards.]
Aleinu leshabeach is the concluding prayer of the services. It contains the phrase "veAnachnu kor’im uMishtachavim" ("and we bend our knees and bow") during which you should bow. This is the knee-waist version of bowing. My siddur and Rabbi Hayim Halevy Donin says you finish at "uMishtachavim", but I have seen other texts that include the following word "uModim" into the bow. In practice it shouldn’t make much of a difference when you pray fast.
That’s it, at least for normal days and Shabat!