My textbook says ある can be used for possession, but further on it says 持っている is used for possession. Are both correct? Are there any differences?
When talking with a Japanese friend (in very limited and simple conversations), when I use ある, she usually corrects me with 持っている.
'Have' in English has many meanings. When you want a more passive sounding 'have', then 「ある」 is best. When you can change 'have' to 'there is...', then use this. 「ある」 can also mean possession, but this is a more passive way of saying it. 「持っている」 is used more for the meaning of 'I have it with me' or 'right here in my hand'. It draws attention to you HAVING it, where 「ある」 places stress on the item existing in a place (that of course may be at your house or in your bag). There is a lot of gray area between the two where both are fine.
If you can reword the English to accept "There is/are..." instead of "...has...," that means use 「ある」.
One example would be:
The station has a cafe. -> There is a cafe in the station.
So the Japanese would be 「駅にカフェがある。」
You can say:
I have a pen on my desk. -> There is a pen on my desk.
(granted your intended meaning is the same). So the Japanese would be 「テーブルの上にペンがある。」
But saying this in English is strange:
I have a pen. -> ?There is a pen (in my hand?).
So here, you would use 「もっている」. If you used 「ある」, it would sound overly indirect.
(As pointed out in comments) This is only a general rule. Whenever you don't want to stress 'possessing an object,' use 「ある」.
「ペンを持っている。」 I have a pen (stressing possession) 「ペンがある。」 I have a pen (not stressing anything, so it is more like 'the pen exists')
In a question: 「ペンある？」 Do you have a pen? (translation depends on context)(natural way to ask) 「ペンを持っている?」 Do you have a pen?
As this last one stresses possession, it may be asking if you have a pen with you, but not necessarily. eg 「３DS持っている。」 which is 'I own a 3DS.' or 'I have a 3DS right now, in my possession.'
「持つ」also has other meanings/uses (「持っていく」 to bring, for example), but that is outside of your question.
Hold contextually. 「ペンを持っている。」will be - I hold a pen with me(same meaning with I have a pen). 「持っていく」will be - Hold( the object) and come(same meaning with to bring).
Carrying 持っている
exists ある
Rough example: You would not say "I am carrying my arm" but you would say "I am carrying an apple"
鉛筆 がある (enpitsu ga aru) means "There is a pencil" and can also mean "I've got a pencil"
ある generally places the emphasis on the fact that the object exists. If you can casually say that something exists, you might have it, if it's something that you can possess. 持っている places the emphasis on the object's presence with you, especially if it's something you're holding with you or with your belongings.
As a result, ある as the verb "to be" can be used to refer to objects and their existence. (i.e. 駅前はカフェーがある。) It can be used to refer to objects you have (i.e. 僕のかばんの中にいろんなえんぴつがある。)
持っている is definitely used for objects that you possess, but I think the contextual limit for using 持っている is something that is portable or movable, such as a car, or a small possession. But, for example, you could not say 持っている about something fixed like a house. You can't carry a house with you.
The background context is that normally, 持っている implies that you carry it with you. Things you carry with you are generally your possessions. But places are not those kinds of possessions, however they do exist. In those cases, using ある is more contextually correct.
