How to compare 2 huge lists of custom objects most efficiently?

I think I can help you.. how do you like this as a teaser? :)

Original Method:   20000 matches, 4197 ms
Dictionary Method: 20000 matches,   27 ms

As long as you "only" want to compare some ten-thousand strings and not billions, you can just use a Dictionary. In a first loop, fill in the "existing" values, then in a second loop look up if a "new" value is found.

I modified your code a bit, but it's essentially the same. Take a look at it and in case you have questions, please leave a comment.

static void Main()
{
    List<string> prejit = new List<string>() { "abc", "def" };

    OriginalMethod(prejit, prejit);
    DictionaryMethod(prejit, prejit);

    List<string> Existing = new List<string>();
    List<string> New = new List<string>();

    for (int i = 0; i < 20000; i++)
    {
        string s = GetRandomString();
        Existing.Add(s);
        New.Add(s);
    }

    for (int i = 0; i < 80000; i++)
        Existing.Add(GetRandomString());

    for (int i = 0; i < 80000; i++)
        New.Add(GetRandomString());

    OriginalMethod(Existing, New);
    DictionaryMethod(Existing, New);

    Console.ReadKey();
}

static string GetRandomString()
{
    string path = System.IO.Path.GetRandomFileName();
    while(!Char.IsLetter(path[0]))
        path = System.IO.Path.GetRandomFileName();
    return path;
}

static void OriginalMethod(List<string> Existing, List<string> New)
{
    var Alphabets = new List<string>() { "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" };

    var sw = System.Diagnostics.Stopwatch.StartNew();

    int matchCount = 0;

    foreach (var character in Alphabets)
    {
        var existingList = Existing.Where(e => e.StartsWith(character)).ToList();
        var newList = New.Where(n => n.StartsWith(character)).ToList();
        foreach (var newItem in newList)
            foreach (var existingItem in existingList)
                if (newItem == existingItem)
                    matchCount++;
    }

    var elapsed = sw.ElapsedMilliseconds;

    Console.WriteLine("Original Method: {0} matches, {1} ms", matchCount, elapsed);
}

static void DictionaryMethod(List<string> Existing, List<string> New)
{
    var existingLookup = new Dictionary<string, List<string>>(Existing.Count);
    List<string> matches;
    int matchCount = 0;

    var sw = System.Diagnostics.Stopwatch.StartNew();

    foreach (string s in Existing)
    {
        if (!existingLookup.TryGetValue(s, out matches))
        {
            matches = new List<string>();
            existingLookup.Add(s, matches);
        }
        matches.Add(s);
    }

    foreach (string s in New)
        if (existingLookup.TryGetValue(s, out matches))
            matchCount += matches.Count;

    var elapsed = sw.ElapsedMilliseconds;

    Console.WriteLine("Dictionary Method: {0} matches, {1} ms", matchCount, elapsed);
}

If all you need is to know whether a matching "existing" value exists or not (and not which one / which ones) you can simplify and increase its "speed" even more by replacing the Dictionary<string, List<string>> by a HashSet<string>.