argument index: This represent which argument goes into the string.

String.Format("first = {0};second = {1}", "apple", "orange");
String.Format("first = {1};second = {0}", "apple", "orange");

"first = apple;second = orange"
"first = orange;second = apple"

alignment (optional): This represent the minimal length of the string.

Postive values, the string argument will be right justified and if the string is not long enough, the string will be padded with spaces on the left.

Negative values, the string argument will be left justied and if the string is not long enough, the string will be padded with spaces on the right.

If this value was not specified, we will default to the length of the string argument.

String.Format("{0,-10}", "apple");      //"apple     "
String.Format("{0,10}", "apple");       //"     apple"

format string (optional): This represent the format code.

Numeric format specifier is available here. (e.g. C, G...etc.)
Datetime format specifier is available here.
Enumeration format specifier is available here.
Custom Numeric format specifier is available here. (e.g. 0. #...etc.) 
Custom formatting is kinda hard to understand. The best way I know how to explain something is via code:

int pos = 10;
int neg = -10;
int bigpos = 123456;
int bigneg = -123456;
int zero = 0;
string strInt = "120ab";

String.Format("{0:00000}", pos);      //"00010"
String.Format("{0:00000}", neg);      //"-00010"
String.Format("{0:00000}", bigpos);   //"123456"
String.Format("{0:00000}", bigneg);   //"-123456"
String.Format("{0:00000}", zero);     //"00000"
String.Format("{0:00000}", strInt);   //"120ab"
String.Format("{0:#####}", pos);      //"10"
String.Format("{0:#####}", neg);      //"-10"
String.Format("{0:#####}", bigpos);   //"123456"
String.Format("{0:#####}", bigneg);   //"-123456"
String.Format("{0:#####}", zero);     //""
String.Format("{0:#####}", strInt);   //"120ab"

String.Format("{0:X00000}", pos);      //"A"
String.Format("{0:X00000}", neg);      //"FFFFFFF6"
String.Format("{0:X#####}", pos);      //"X10"
String.Format("{0:X#####}", neg);      //"-X10"

The "0" specifier works well with other numeric specifier, but the "#" doesn't. Umm... I think the "Custom Numeric Format String" probably deserve a whole post of it's own. Since this is only the "101" post, I'll move on to the next argument in the format string.

zeros (optional): It actually has a different meaning depending on which numeric specifier you use.
int neg = -10;
int pos = 10;
// C or c (Currency): It represent how many decimal place of zeros to show.
String.Format("{0:C4}", pos);      //"$10.0000"
String.Format("{0:C4}", neg);      //"($10.0000)"

// D or d (Decimal): It represent leading zeros
String.Format("{0:D4}", pos);      //"0010"
String.Format("{0:D4}", neg);      //"-0010"

// E or e (Exponential): It represent how many decimal places of zeros to show.
String.Format("{0:E4}", pos);      //"1.0000E+001"
String.Format("{0:E4}", neg);      //"-1.0000E+001"

// F or f (Fixed-point): It represent how many decimal places of zeros to show.
String.Format("{0:F4}", pos);      //"10.0000"
String.Format("{0:F4}", neg);      //"-10.0000"

// G or g (General): This does nothing
String.Format("{0:G4}", pos);      //"10"
String.Format("{0:G4}", neg);      //"-10"

// N or n (Number): It represent how many decimal places of zeros to show.
String.Format("{0:N4}", pos);      //"10"
String.Format("{0:N4}", neg);      //"-10"