1
We are נוהג to get up באשמורת to say סליחות and תחנונים from ראש חודש אלול and on until יום כפור [1]. This is because these days are ימי רצון during which משה רבינו ascended הר סיני to receive the second לוחות [2]. So is the מנהג of the ספרדים. The מנהג of אשכנזים, however, is that where ראש השנה is on Thursday or שבת then we start saying סליחות on the Sunday immediately prior. (Some also fast and daven a תפלת תענית, including עננו and ויחל, etc.) [3]. The חשבון of these 4 days before ראש השנה on which סליחות are said is to compensate for the 4 days between ראש השנה and יו"כ on which we eat סעודות מצוה such that we cannot fast as some used to [4]. Where ראש השנה falls on Monday or Tuesday, however, then we start on Sunday of the week before ראש השנה, rather than the Sunday of the week of ראש השנה [3]. The reason for this difference is because even though, strictly speaking, 4 days would be enough, nevertheless they instituted that סליחות should consistently begin on Sunday, so that no one should get confused what day סליחות start [5]. אשכנזים do, however, blow שופר every morning (and in some places every evening) from ראש חודש and on, which is a subject for its own article.
The reason for the specification of אשמורת is because הקב"ה floats, as it were, in 18,000 metaphysical worlds, and at the end of the night (which is, evidently, what אשמורת refers to) - He floats, as it were, in this world, such that the end of the night - i.e. before dawn - is an עת רצון [6]. The source of this concept (as pointed out by ביהגר"א [7]) is איכה ב:יט where the idea is expressed that ראש אשמרות (lit. the head of watches) is an appropriate point at which to pour one's heart out to Hashem.
2
There are those who are נוהג that whoever leads סליחות also leads the davening throughout the rest of the day [8].
3
On a practical level, ערוה"ש [9] offers an advisory that those who are not fasting should replace words referencing fasting in a given סליחה with words referencing תפלה. Additionally, there are words in סליחות referencing night or אשמורת הבוקר as the time where the particular סליחה is being said. One who says סליחות in the morning (i.e. before שחרית) should skip those words so as not to be saying falsehood.
Postscript:
There are various other technical דינים relating to who should lead סליחות, in what specific order they need to be said with what liturgical framework (such as אשרי, קדיש and the like), what do if one is wants to say סליחות individually, etc. These are purely halachic (rather than מנהג) in nature and, as such, are far beyond the scope of this article.
[1] שו"ע או"ח תקפא:א
The טז (ס"ק ב) points out that on ר"ח itself, however, we do not say סליחות and תחנונים.
[2] ביהגר"א אות א שם ע"פ פרקי דרבי אליעזר פרק מו
[3] טור שם
[4] פרישה שם ס"ק ו
[5] שם ס"ק ז
[6] מ"א שם ס"ק א
A vague reference to this point is made in :עבודה זרה ג.
[7] שם אות ב ובפסוק "קומי רוני בלילה לראש אשמרות שפכי כמים לבך נכח פני ה" ופי' הריד שהוא לשון תפלה, כמו אל הרנה ואל התפלה
[8] רמא שם
[9] שם סעיף ד