Learn Swedish: Postombudet räcker kunden avin. - The postal agent hands the customer the notification.

Just like in English, when you don't use a preposition like 'to' (till), the person receiving the item (the indirect object) comes immediately after the verb. So kunden (the customer) is the receiver, and avin (the notification) is the item being handed over.

How would this sentence change if I wanted to use the word till (to)?

If you use a preposition like till, the item being handed over (the direct object) comes first. The sentence would become: Postombudet räcker avin till kunden (The postal agent hands the notification to the customer).

I thought the verb räcker meant 'to be enough'. Why is it translated as 'hands' here?

You're right that räcka often means 'to be enough' (like in the phrase det räcker, 'that is enough'). However, when it is used with an object to physically transfer something, it means 'to hand' or 'to pass' something to someone.

What exactly is avin, and what is its base form?

The base word is en avi, which refers to a paper slip or notification, commonly used by the Swedish post office to tell you a package has arrived. Avin is the definite form ('the notification'). Since the root word ends in a vowel, it just takes an -n at the end to become definite instead of -en.