Translation:My finger is smaller than your finger.
Think of the pronoun अपना as being '----'s own' where the ---- stands for the subject of the sentence.
When you have a sentence like 'I am at my grandmother's house', the pronoun 'my' refers to the subject of the sentence which is 'I'. So, it is replaced with अपना - मैं अपनी दादी के घर में हूँ।
Similarly, in 'Aamir is doing his (own) work' and 'You are going from my house to your house', the pronouns 'his' and 'your' refer to the subjects of the respective sentences, 'Aamir' and 'you'. So, the translations would be 'आमिर अपना काम कर रहा है।' and 'तुम मेरे घर से अपने घर जा रहे हो। '
As you can see the meaning of अपना changes depending on what the subject of the sentence is. In contrast, तुम्हारा is always 'your' as are the less formal version तेरा and the more formal version आपका.