Prayer brings us closer to God.  When we ask for things from God, even though God already knows what we want and need, we get merit (zechut) for the prayer.
Particularly in synagogue, a holy place, we must be always cognizant of God's presence and take special care to show respect to God in His house. This awareness is even more important than memorizing the halachot of prayer: those who speak in synagogue are showing disrespect to God as well as disturbing others and preventing them from praying with concentration (kavana).
The three daily prayer services--shacharit, mincha, and ma'ariv--are related to the three forefathers who instituted them.  They partly take the place of—and are modeled after--the Temple sacrifices.
If you find you have made an error in saying a blessing or prayer, you may correct your error without having to repeat any previous parts if you do so within 2.5 seconds.

The minimum amount of prayer that women and girls 12 years old and up must say varies according to several opinions:

• Opinion that RMH approves of:
  • Birchot ha'shachar (from al netilat yadaim--to u'maflee la'asot)
  • Birchot haTorah (from la'asok b'divrei Torah--to notein haTorah)
  • Yevarechecha/Eilu divarim she'ein/Eilu divarim she'adam
  • Elohai neshama
  • Asher natan 'sechvi... gomel chasadim tovim l'amo Yisrael
  • Shema (first line of shema plus Baruch shem kevod malchuto l'olam va'ed)
  • If they choose to say the morning amida, they should precede it with emet, v'yatziv until ga'al yisrael.
• Ramban: Women should say at least the amida of shacharit and of mincha.
• Another opinion: Women must say some prayer every day (but it may be any type of prayer, not only the fixed prayer services).
• Another opinion: Women do not need to say the amida and emet v'yatziv on Shabbat and Jewish festivals (that is, whenever we are not asking for anything). 
• Women who have small children who require the mother's constant attention should not say any prayers except for birchot ha'shachar and birchot haTorah.

For more on women and the shema, see Women and Shema in Shacharit.

Introduction to Missed Amida/Tashlumim
If you accidentally miss any amida, you may make up for it (tashlumin)--unless you intentionally missed it.

Amida: Errors: Missed Amida/Tashlumin: Regular Days
Missed Ma'ariv Amida
If you accidentally did not say the amida for ma'ariv:

• Say the normal shacharit amida the next morning with the other men in the minyan.
• Say ashrei.
• When the leader begins his repetition of the amida, say the amida along with him, word for word, including kedusha.
• After saying ha'el ha'kadosh, finish your amida at your own pace.
• If you are not with a minyan, say ashrei and then repeat the shacharit amida.

Missed Shacharit Amida
If you accidentally did not say the amida for shacharit:

• Say the normal mincha amida with the other men in the minyan.
• Say ashrei.

• When the leader begins his repetition of the amida, say the amida along with him, word for word, including kedusha.
• After saying ha'el ha'kadosh, finish your amida at your own pace.
• If you are not with a minyan, say ashrei and then say the mincha amida a second time.

Missed Mincha Amida
If you accidentally did not say the amida for mincha:

• Say the normal ma'ariv amida with the other men in the minyan.
• Say ashrei.
• Say the ma'ariv amida a second time. Skip modim.
• If you are not with a minyan, say ashrei a second time and repeat the ma'ariv amida.

Note At the next prayer service, say whatever is the correct amida for that later prayer service, even if it is not the same amida that you missed.
If you miss mincha on Friday, say the ma'ariv service for Shabbat and repeat that amida again.
Note Once the time for the next amida has passed, you may not make up the missed amida.
Example
If you missed mincha on Thursday, you may only say tashlumin for mincha as long as you may still say ma'ariv, which is daybreak of Friday morning.

Amida: Errors: Missed Amida/Tashlumin: Shabbat/Jewish Festivals
Even if you forgot to say a prayer service on Shabbat and Jewish festivals, say the next prayer service amida and repeat THAT amida to make up (tashlumin) for the one you missed--even if it is no longer Shabbat or the Jewish festival.
Exception
There is no tashlumin for musaf. However, you may say musaf until sunset, even if you already said mincha.
Note If the time for mincha has arrived (½ hour after halachic midday), you must say mincha before saying musaf (but if you could join a mincha minyan later, you may say musaf now).
Situation
You miss Shabbat mincha.
What to Do
Say ata chonantanu in ma'ariv, but only for the first time you say the amida, not the second time (which is tashlumin).
Situation
You forget to say ma'ariv on Saturday night.
What to Do
Say ata chonantanu on Sunday morning in the second amida (which is tashlumin).

If Doubt about Whether You Said Amida
If you are not certain whether you said an amida:

• On a weekday, assume that you did not say the amida and say it anyway.  Intend that:
  • If you forgot the previous amida, this makes up for it, and
  • If you did say the previous amida, the second one is a nedava (free-will “offering”).
• On Shabbat and Jewish festivals, do not say the amida twice as tashlumin.

            Reason You may not say a voluntary prayer service (nedava) on these days.
Men who intentionally delayed saying shacharit past the fourth halachic hour of the day still say that amida until midday but if they did not say it by midday, they may not say tashlumin. See Minyan: Keeping Pace: Shacharit Minyan.

Introduction to Fast Days
The purpose of Jewish fast days is to make us reflect on our behavior and improve it by:

• Eliminating negative actions and thoughts, and
• Taking more care in our observance of the commandments.

God told the Jewish people on many occasions that He did not want their pointless sacrifices or fast days but rather wanted the Jews to improve their behavior. If people fast but still have reprehensible behavior, the people have missed the point and purpose of bringing sacrifices and having fast days.
All fast days may be circumvented for health reasons. Consult a rabbi.
Fast days come in several varieties:

• Major fast days: Yom Kippur and the Ninth of Av (Tish'a B'Av). 
• Minor fast days: There are four minor fast days; see below.
• Fast for First-Born
• Individual Fasts

The Major Fasts
Yom Kippur and the Ninth of Av begin before sunset and are both slightly longer than 24 hours. Yom Kippur is a festive day while, on the Ninth of Av, Jews mourn for the destruction of both Jerusalem Temples (the first was destroyed by Babylonians; the second, by the Romans). On both fasts, it is forbidden to:

• Eat or drink.
• Wear leather shoes (but you may wear any other clothing made of leather).
• Bathe (or even to wash anything more than fingers; you may not use a deodorant--not even a spray type--since it is similar to washing).
• Have marital relations.
• Use cosmetics or body scents.

The Minor Fasts
The four minor fasts begin 72 minutes before local sunrise. They technically end at dark but functionally end after ma'ariv (and for the Fast of Esther, after hearing the megila). Unlike most times in the Jewish calendar, this 72 minutes is normal time and NOT based on the current length of the day (sha'a zmanit):

• Fast of Esther (Ta'anit Esther): Day preceding Purim (or sometimes earlier).
• Fast of Gedalia (Tzom Gedalia): Usually the day after Rosh Hashana.
• Tenth of Tevet (Asara b' Tevet):
• Seventeenth of Tamuz (Shiv'asar b'Tamuz): Beginning of the Three Weeks of semi-mourning beforeTish'a b'Av.

On all fast days, eating and drinking are forbidden. The Tenth of Tevet and the Seventeenth of Tamuz additionally have the same restrictions as the Nine Days. So you may wash or bathe on the Fast of Esther and on Tzom Gedalia but not on the Tenth of Tevet and the Seventeenth of Tamuz. (See Nine Days: Restrictions).
Note On a fast day that has been delayed for Shabbat, there may be leniencies for:

• The mohel, sandak, and father of a baby getting a brit mila to eat after mincha.
• Pregnant or nursing women (this leniency applies to eating throughout the day).

Fast for First Born
Fast for first-born males 13 years old and older, on the day before Passover. This includes first-born cohanim and Levites (levi'im). The fast begins 72 normal minutes before sunrise and ends with kiddush at the seder. However, the widespread custom is to attend a siyum on the day before the first seder so that the first-born men do not need to fast on that day.
 
Personal Fasts
Fast by a groom and bride on their wedding day is an example of personal fasts. This fast begins 72 normal minutes before sunrise and ends with drinking wine under the chuppa.