AES plaintext is smaller than 128 bits - how to expand?
A block cipher is an invertible transformation that maps an $n$ bit block of bits to an $n$ bit block of bits, under the control of a key (and where $n=128$ in the case of AES)

Now, we most often need to do things other than mapping blocks of $n$ bits; how we do that is using the block cipher within a Mode of Operation.  A mode of operation is just a way to use the block cipher to solve some problem we want solving.

There have been quite a number of modes of operations defined; the most common ones are CBC and CTR (counter).  The majority of them (including CBC and CTR) solve the problem 'how do I encrypt an arbitrarily sized message'.

As for 'how to handle odd sized messages', CBC and CTR take two different approaches.

With standard CBC, we assume that the message is always a multiple of $n$ bits; so, what we do is add padding to the message before we do that actual CBC-mode transformation.  This padding fills out the message to a multiple of $n$ bits; this padding is designed so that the decryptor can easily remove it after decryption.  Note: I said standard CBC because there are fancier ways to avoid this padding within CBC mode; they involve handling the last partial block as a special case.

With CTR mode, it doesn't assume that the message is a multiple of $n$ bits at all.  It doesn't actually send the message through the block cipher; instead, it generates a keystream by sending an incrementing pattern through the block cipher (so the first $n$ bits of the keystream may be generated by encrypting the value 1, and the second $n$ bits of the keystream may be generated by encryption the value 2, etc).  One we have such a key stream, having the message not being a multiple of $n$ bits is not an issue; we just discard the parts of the keystream we don't need to encrypt.