I've been asked by no one less than the Global Moderator herself to shed some light on this because in my italki profile, I say that I teach both Standard and Common Czech. So what's the difference?
Standard Czech is the codified, "proper" language used in media, formal documents, non-fiction literature etc. It was standardised and codified during the Czech National Revival in the late 18th - early 19th century. It's called National Revival because we were so heavily Germanised at the time and the Czech language had been almost entirely eradicated from state administration, schools, literature etc.
Common Czech (sometimes erroneously referred to as Colloquial Czech) is not just a set of colloquialisms. It is an actual dialect (interdialect really) that is spoken in Bohemia, the western half of the Czech Republic, including the capital, Prague. It is not codified but its "rules" or tendencies have been described in literature and are fairly consistent. It is how people speak absolutely everywhere within the area, unless in a very formal situation. If Standard Czech were used in an informal setting, say at a pub in Prague, it would be considered awkward or inappropriate. The difference between Standard and Common Czech is quite big, and is mostly of morphological or phonological nature. Certain letters are added or omitted and certain suffixes (word endings) change, depending on gender, number and grammatical case. Standard Czech: "s mladými lidmi" changes to "s mladejma lidma" ("with young people"). The real colloquial language can affect whole words ("dům" vs "barák" = a house).
The problem is that the eastern part of the country happens to really like Standard Czech - they still have dialects there but very different to Common Czech and I'd say more subtle - and often really dislike Common Czech. They use Standard Czech even in informal settings without it sounding awkward. Teachers who come from those areas will automatically teach their students Standard Czech which is fine if their students visit the same town they live in. If they go to Prague without any knowledge of Common Czech, they will be lost. Not only will they sound awkward, they will not understand what others are saying. That's why I explain this to my students and let them decide, whether they want Standard Czech only or both. It usually depends on whether they're actually planning on visiting the country or not. Those who already live here, usually ask for "Prague" Czech automatically.
It's a very unique situation and it cannot be compared to how countries with multiple small dialects function. Having one "Common" language that half (possibly even more) of the population speak 90% of the time is something worth mentioning to potential students, in my opinion.