Calibre is a software, which you can install on your computer, not on your reader.
You can organize your books with it, convert them from almost any format to almost any format there is.
By installing some extra plugins, you can also use it to free your purchased books from DRM-lockin (google Apprentice Alf), so you are able to buy from practically all book vendors, regardless of the format.
Regarding the dictionary issue:
.mobi-format has an implementation, but it is really getting more and more out of fashion. Except for Kindle, all readers are going epub, so mobi dictionaries are of no use there.
The only reader coming to my mind, which still supports mobi lookup is the Cybook Opus with mobi firmware. No touch there. I used a Cybook Gen3 some time ago (practically identical firmware) and was very happy with the PONS dictionary I could use then. Just the lookup procedure was a bit tedious.
I'm not sure, but I think, on a Kindle you can also buy and install your own dictionaries.
Many epub-readers now come with preinstalled dictionaries. If you are happy, with what they offer (eg. Sony), you have a wide variety of readers. If you want to get the dictionaries of your choice onto your reader, see above.
You can also checkout the famous
wiki reader matrix.
edit: I may be wrong, but I am not aware of the possibility of using mobi dictionaries while reading pdf. Generally pdf is the worst format to use with an ereader.