という

Meaning

A X called Y
(Noun)YというX

という is a combination of quotation と and 言う (to say). It is a phrase that defines what something is called, i.e. it’s name.
It is commonly used when mentioning the name of something that you think the other person won’t know. 
Although this is a combination ofと and 言う, it is usually written in かな. 

Examples

• バラという花 – A flower called “rose”
• 玉木亭というパン屋 – A bakery called “Tamakitei”.
• ジョンという男 – A man called Jon.
• 黄金という金色の金属 – A gold colored metal called おうごん (gold). 

VS の

Sometimes I get asked how this is different to の.
Although this is a noun modifying a noun, it is very different from の:

• タンポポのはな – Dandelion flower
• タンポポという花 – A flower called a dandelion.
• ジョンの男 – Jon‘s man.
• ジョンという男 – A man called John.

Sentence Nominalization

という also serves as a way to make a sentence modify a noun.
This is basically just an application of verb modification and the quotation と particle.
Typically, this kind of sentence modification is used with certain words in particular, such as:
こと、くせ、もの、説, and 話

See also ということだ

Examples

• 猫という動物 – An animal called a cat.
• 山という岩 – A boulder called a mountain.
• 人を無視するというくせがある – He has a habit of (called) ignoring people. 
• 彼がクマに襲われて、素手で倒したという話聞いた？
  Did you hear the story of how he was attacked by a bear and beat it with his bare hands?
• 火山で作られたという説がある – There is an explanation of them being created by volcanoes.

とかいう

A X Probably called Y

とか is a less certain version of と, which can be used with という as とかいう。
This changes the meaning from “Called” to “Probably called”; or “I think it was called”.

Examples

• ブライアンとかいう人に会った – I met a guy (probably) called Brian.
• ロギンとかいう店で飲んだ – I went a store called Rogin (probably). 
• スフィアとかいう猫を飼っている – They have a cat called Sphia? I think. 
  
  

って

って
と or という

という is frequently abbreviated to って。
と may also be abbreviated this way when used as a quotation with words like いう or 思う.
という is a little special in that the いう also tends to be dropped.

• 玉木亭って店 – A store called Tamakitei
• ジョンって男 – A man called Jon.
• ピザって食べ物 – Food called pizza.