With specifying extras=url_o you get a link to the original image:
https://api.flickr.com/services/rest/?method=flickr.photos.search&api_key=YOURAPIKEY&format=json&nojsoncallback=1&text=cats&extras=url_o
For downscaled images, you use the following parameters: url_t, url_s, url_q, url_m, url_n, url_z, url_c, url_l
Alternatively, you can construct the URL as described:
http://farm{farm-id}.staticflickr.com/{server-id}/{id}_{secret}.jpg
or
http://farm{farm-id}.staticflickr.com/{server-id}/{id}_{secret}_[mstzb].jpg