It is one of the letters a through z (upper or lower case) or one of the integer digits 0 through 9,

In[1]:= CharacterRange[" ", "~"]

Out[1]= {" ", "!", "\"", "#", "$", "%", "&", "'", "(", ")", "*", "+", ",", "-", ".", \
"/", "0", "1", "2", "3", "4", "5", "6", "7", "8", "9", ":", ";", "<", "=", \
">", "?", "@", "A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", \
"M", "N", "O", "P", "Q", "R", "S", "T", "U", "V", "W", "X", "Y", "Z", "[", \
"\\", "]", "^", "_", "`", "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", \
"k", "l", "m", "n", "o", "p", "q", "r", "s", "t", "u", "v", "w", "x", "y", \
"z", "{", "|", "}", "~"}

Those of these which match WordCharacter:

In[2]:= Select[CharacterRange[" ", "~"], StringMatchQ[#, WordCharacter] &]

Out[2]= {"0", "1", "2", "3", "4", "5", "6", "7", "8", "9", "A", "B", "C", "D", "E", \
"F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", \
"U", "V", "W", "X", "Y", "Z", "a", "b", "c", "d", "e", "f", "g", "h", "i", \
"j", "k", "l", "m", "n", "o", "p", "q", "r", "s", "t", "u", "v", "w", "x", \
"y", "z"}

Greek characters are not part of this:

In[3]:= Select[CharacterRange["\[CapitalAlpha]", "\[CapitalOmega]"], 
 StringMatchQ[#, WordCharacter] &]

Out[3]= {}

WordCharacter represents a single word character. So the string pattern

WordCharacter..

represents a sequence of one or more word characters. So your StringCount computation,

StringCount["Hello my name is Kyle",WordCharacter..]

is returning the "lumps" that correspond to full self-contained groups of word characters. This is what StringCount is designed to do.

?StringCount

StringCount["string","sub"] gives a count of the number of times "sub" appears as a substring of "string". 
StringCount["string",patt] gives the number of substrings in "string" that match the general string expression patt. 
StringCount["string",{Subscript[patt, 1],Subscript[patt, 2],\[Ellipsis]}] counts the number of occurrences of any of the Subscript[patt, i]. 
StringCount[{Subscript[s, 1],Subscript[s, 2],\[Ellipsis]},p] gives the list of results for each of the Subscript[s, i].  >>

For fun compare your example with

StringCount["Hel8lo my name is Dav7id", LetterCharacter ..]

The fact that this is either a documentation error or a bug was noted on Mathematica StackExchange:

http://mathematica.stackexchange.com/questions/24077/how-to-count-words-in-a-greek-text/24080#24080

Thank you very much. I have learned quite a bit through this discussion.

Thank you very much again for your help.