die Tasche
bag
A container made of flexible material with handles or straps for carrying objects, e.g. a handbag, shopping bag, or travel bag.
Sie kramte in ihrer Tasche nach dem Schlüssel.
She rummaged in her bag for the key.

bag / pocket
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bag
A container made of flexible material with handles or straps for carrying objects, e.g. a handbag, shopping bag, or travel bag.
Sie kramte in ihrer Tasche nach dem Schlüssel.
She rummaged in her bag for the key.

'Tasche' (bag/pocket) is a very productive base word for many compounds. Depending on the first part of the word, the type of bag or pocket is specified more precisely, for example: 'Handtasche' (handbag), 'Hosentasche' (pants pocket), 'Jackentasche' (jacket pocket), or 'Einkaufstasche' (shopping bag).
Er hat sein Portemonnaie in der Jackentasche vergessen.
He forgot his wallet in his jacket pocket.
The word 'Tasche' (bag/pocket) is often used figuratively in idioms related to money. When verbs like 'greifen' (to reach), 'liegen' (to lie), or 'wirtschaften' (to manage/economize) appear in a financial context, a figurative meaning is usually intended.
Für das Luxusauto musste er tief in die Tasche greifen.
He had to dig deep for the luxury car.
Pay attention to the case with the preposition 'in'. 'In die Tasche' (into the bag/pocket, accusative) describes movement, while 'in der Tasche' (in the bag/pocket, dative) describes a location.
Ich stecke mein Handy in die Tasche.
I put my phone into the bag.
The idiom 'tief in die Tasche greifen' (to dig deep) is fixed. The verb is 'greifen' (to reach), and the person who pays a lot is the subject of the sentence. The expression is often combined with 'müssen' (must/have to).
Wer ein Haus bauen will, muss tief in die Tasche greifen.
Anyone who wants to build a house has to dig deep.
In the expression 'jemandem auf der Tasche liegen' (to live off someone), the person who is financially burdened is in the dative. The subject is the person who costs the money.
Er ist schon 30 und liegt immer noch seinen Eltern auf der Tasche.
He is already 30 and is still living off his parents.
The idiom 'in die eigene Tasche wirtschaften' (to line one's own pockets) is a fixed unit. The adjective 'eigene' (own) is adjusted to match the subject.
Man warf ihm vor, jahrelang in die eigene Tasche gewirtschaftet zu haben.
He was accused of having lined his own pockets for years.
subject / compartment
'die Tasche' and 'das Fach' can both be small spaces where you keep things. In this context, 'Tasche' is part of a piece of clothing, for example on a pair of pants or a jacket. A 'Fach' is a separate compartment in a cupboard, shelf, backpack, or drawer. The words can be confusing because both provide space for objects. In practical terms: for clothing, you usually say 'Tasche'; for furniture or containers, 'Fach' is more common.

Zu Hause steckt der Gast seinen Schlüssel zur Sicherheit in die Jackentasche.
At home, the guest puts his key in his jacket pocket for safekeeping.
Morgen legt meine Freundin die Quittung in das kleine Fach ihrer Geldbörse.
Tomorrow my friend will put the receipt in the small compartment of her wallet.