What is the breed of the dog...?

Parsons Russell and Jack Russell are the same breed of dog.

Maybe it's the Rat Terrier you are think of. This is a Rat Terrier

http://static.gotpetsonline.com/pictures-gallery/d...

ADD:

It doesn't look like a Jack/Parson Terrier at all. Smooth Fox Terrier looks similar to the Jack/Parson Russell. Personally, it looks like a Brittany Spaniel.

On 1 August 1999, the PJRTC (Parson Jack Russell Terrier Club) successfully petitioned the Kennel Club (UK) to change the name of the breed to the Parson Russell Terrier, with the name of the breed club following suit.[1] The international kennel association, the Fédération Cynologique Internationale, recognised the Parson Russell Terrier on 4 June 2001.[12] The American Kennel Club updated the name of the recognised breed from Jack Russell Terrier on 1 April 2003.[11] The United Kennel Club adopted the new name on 23 April 2008.[9]

In America, it was first recognised as the Jack Russell Terrier in 1997. The name was changed to its current form in 1999 in the UK and by 2008 all international kennel clubs recognised it under the new name.

Other names Parson

Parson Jack Russell Terrier

Jack Russell Terrier (AKC 1997–2003)

http://en.wikipedia.org/wiki/Parson_Russell_Terrie...

Smooth Fox Terrier:

http://www.alldogpictures.tk/smooth-fox-terrier/sm...

Brittany Spaniel:

http://cdn-www.dailypuppy.com/media/dogs/anonymous...

A Parson Russell Terrie is not the same as a jack Russel Terrier. Since many terrier breeds can be differently marked (brown nose and brown spots) it's difficult to tell which "other" dog you are referring to. Could be a Rat Terrier, or a [toy/wire/smooth] Fox Terrier, Brazilian Terrier....