The very weird word counter
Python 2, 220 ... 124 119 Bytes

Huge thanks to Sp3000 for saving a lot of bytes.



f='qwertyuiopasdfghjkl zxcvbnm'.find
g=lambda i:sum(abs(f(x)%10-f(y)%10)+2*abs(f(x)/10-f(y)/10)for x,y in zip(i,i[1:]))


Usage:

g("tttt") -&gt; 0


Check it out here.

Slightly ungolfed + explanation:

f='qwertyuiopasdfghjkl zxcvbnm'.find  # Defining keyboard rows and aliasing find
g=lambda i:                           # Defining a function g which takes variable i
    sum(                              # Sum of 
        abs(f(x)%10-f(y)%10)          # horizontal distance, and
        + 2*abs(f(x)/10-f(y)/10)      # vertical distance,
        for x,y in zip(i,i[1:]))      # for each pair in the zipped list

# Example of zipping for those unaware:
# Let i = asksis, therefore i[1:] = sksis, and zip would make
# the list of pairs [(a,s),(s,k),(k,s),(s,i),(i,s)].