When you place an order at Aliexpress online marketplace, the merchant prepares your goods for shipping with a certain carrier. Which one will it be, depends on your choice, since you have several options. If you choose the free delivery, then your package will be most certainly handled by China Post Small Packet. You also have other options, like DHL Express, but it will cost you more.
As usual, you will be provided a tracking number as soon as your order is shipped. You have to find the order you want to track at "My Orders" section, then go to the "Details" page of that order, then find and copy the tracking number.
The way to find out the latest information about your package from Aliexpress is to insert your tracking number into a form at the top of this page and click the "Track" button.
Also, don't confuse your tracking number with Aliexpress order number, those are completely different things. The order number is just an internal system ID, which usually consists of 14-15 digits, e.g. 492056398564738. You cannot track your parcel with that number. On the contrary, tracking numbers usually contain letters, and even if they don't, they are usually shorter, than order number, e.g. 375584930274, LY928374872CN, etc.