How do I put my logo on a product?

Short answer: On a shop draft, click the product in its collection: opposite each placement, click the thumbnail of the logo you want to apply, then select its version (the size) in the drop-down menu just underneath.

Steps

Open the “Shops” page and select your shop.
A live shop cannot be edited directly. Click the orange “Modify the shop (create a draft)” button and work on the draft. If you are already on a draft, skip this step.
In a collection, click the product. Its configuration opens.
The configuration lists every placement available on that product (“Left chest logo”, “Logo on the chest”, “Logo at the top of the back”…). Opposite the placement you want, a row of thumbnails offers “None”, then each of your logos: click the thumbnail of the logo you want to apply.
Just under the thumbnail, a drop-down menu gives the version (the size) and its price — see How do I select the logo size on a product.
Repeat for every placement you want, then save.

The product needs at least one non-optional logo: that is the one that sets its price. Otherwise the system refuses with “Please select at least one non-optional logo”.

On Kids / Junior products, use the Junior version, otherwise the logo may not fit on the product when it is applied. See What is the Junior version for.

Some placements are mutually exclusive: as soon as you switch one on, the incompatible placements are grayed out. See Where can I place my logo on a product.

Placements are chosen on the first color

The placements you switch on apply to every color of the product, and they can only be set from the first color. On the other colors, the only thing you can still change is the logo itself — which is exactly what you need to do when you have one logo for a light background and one for a deep background.

Going further

Where can I place my logo on a product — the available placements and which ones exclude each other.
Can I use different logos on different products
Can I offer the buyer a choice between several logos

You'll know it worked when: The product shows the logo in its visual and its price includes the marking.