The nature of assets
The elements of financial statements are the building blocks of financial statements. If your students understand what the elements are then they have a good basis for understanding how events are reported in financial statements. Those elements – assets, liabilities, equity, income and expenses – are key to understanding financial reporting. Something must meet the definition of one or more of those elements to appear in the primary financial statements. 
The nature of assets is a good place to start, in the context of teaching accounting concepts, because they are valuable. Of the five elements of financial statements only assets are valuable. This is an under-appreciated point and can lead, in later lessons, to the dramatic claims that ‘equity is not valuable’ and ‘income is not valuable’.
‘Asset’ is the accounting label that we use to describe the resources of a business: resources that are used to generate profits. Some assets are monetary, like cash, and have an obvious value. Some assets are non-monetary, like brands or buildings, and the figure that accompanies them is our estimate of their value to us.
The IFRS Conceptual Framework  defines an asset as:
a present economic resource controlled by the entity as a result of past events
And an economic resource is:
a right that has the potential to produce economic benefit
Here are some suggestions for teaching the nature of assets, using different aspects of the definition from the Conceptual Framework.
Teach resources first
Assets are resources first of all. So often I start by asking students to identify what resources a business might have, maybe using a photo of a business, like the inside of a shop. They will suggest the usual things like the building, fixtures, inventory, maybe brand names, and so on.
Keep going long enough and you’ll get most things that could be an asset. Someone will suggest people, at which point you will probably want say ‘no’ but that’s fine for the moment. People are an important resource for a business.
Assets are rights, not things
The next key idea to get across to students is that the asset is never the object itself. It is the right to make profits from the object.
A building is not an asset. Because an empty building is not profitable. Your (exclusive) right to use the building, to sell things from, or to rent out to tenants, is the asset, because you make profits by using the building. This is why we might value our building using two different bases (cost or market value) depending on how we intend to use it.
When you buy a train ticket, you have the right to travel from one place to another. The value of the train ticket lies in the benefits you get from making that journey – to go to work and earn money, or to see a client and charge them for your time. It’s not the intrinsic value in the piece of cardboard on which the ticket is printed. 
Once the ticket expires, it has no value at all even though it’s the same piece of cardboard it was before. See this lesson resource which explains the idea in more detail.
Recognition: not all resources are recognised as assets
Assets are resources but not all resources are recognised as assets. For an asset to be recognised it must meet certain criteria, i.e., control, and the potential to produce economic benefits. And sometimes additional criteria specific to the nature of the asset, for example, a reliable measurement of its value.
Financial statements have to give relevant information – which implies that resources whose value is negligible, either by their nature or their expected life, might conceptually be identical to assets but won’t appear on the statement of financial position.
So for example, if you hire a car for four years, and you have to service and maintain that car, you will recognise an asset, which is then carefully measured and remeasured at every reporting date.
But you won’t recognise an asset if you hire a car for a week to tour around on holiday. Not because the two cars are conceptually different but because in the second case the benefits of providing the information to users of financial statements don’t outweigh the costs of doing so. 
Control is what links an economic resource to a business.
For example, a retailer does not control the rights associated with ownership of the entire shopping centre or the land on which it is built. But it does control the economic rights associated with having exclusive use of a retail unit within a shopping centre. It is these specific rights that must be recognised as assets.
If you have control over rights, that means that you will enjoy the future economic benefits of the asset. So, in general, if you can direct how the resource is used, and stop anyone else from using it, then you have control.
You have the right to walk down a public street, but you wouldn’t recognise an asset in your financial statements. You don’t have the right to stop anyone else from doing so and deriving the same benefits. So you can’t demonstrate control.
Running a business is all about getting the most out of its resources. If your students understand the nature of assets, they will understand business as well as accounting.
Notes, references and further reading
[Note 1] When it suits. Deferred tax stretches it a bit. And if you really want to live dangerously, point your students towards hedges of unrecognised firm commitments in IFRS 9 Financial Instruments and ask them about recognising an asset in respect of a movement in an unrecognised liability.
[Note 2] Conceptual Framework for Financial Reporting paras 4.3 – 4.4 (IASB, 2018)
[Note 3] I know that you can get train tickets on your phone now.
[Note 4] IFRS 16 Leases para 5 (IASB, 2020)