Path pasted from Illustrator no stroke

Copying a vector path from Illustrator and pasting it in After Effects does not retain any of the properties except the path. For the position to be the same the artboard in Illustrator must be the same frame size in points or pixels as the comp in After Effects.

If you import a layered Illustrator file as a composition retaining layer size, then any path or shape in Illustrator can be converted into a shape layer. Gradients and brushes, and most effects you apply in AI will also not successfully convert to shape layers. You'll end up with a mess that can be difficult to sort out. The only reason you would ever want to convert a vector file (AI) to a shape layer is to use Shape Layer animators, animate the path, or extrude the shape layer using the C4D rendering engine. 

If you have an empty path on a shape layer and you select the path and click on the Add menu button at the top right corner of an expanded shape layer's Switches or Modes column in the timeline and select Stroke, it will add a stroke to that path. You can even have more than one stroke on a single path.

To see all modified properties on a shape layer, select the layer, press the 'u' key twice, and presto; all modified properties of the layer appear. That's what I did to take that screenshot.

If you copy a path in Illustrator then switch to After Effects, select a shape layer or add a new shape layer, then select the pen tool and set stroke color and width, then click anywhere in the comp panel to start the path, then paste, the path from AI will have the stroke and/or Fill you selected with the pen tool. If you just select the shape layer, the path will not have a stroke or fill.

I hope this helps. If you are new I strongly suggest that you become familiar with the User Guide and check out a bunch of the projects there to learn the UI and basic workflow. Use the Search Help field in the top right corner to look up specific subjects. You can also get a good idea of how the UI works and how to customize your workspace and reset it by just opening the Learn workspace or clicking on the Learn link on the After Effects Home screen. Be warry of tutorials you find on YouTube. Most of them are prepared by enthusiasts that have a limited ability to clearly explain workflows, and some of them are just rehashed recipes from other enthusiasts. Vet your trainers. The number of views is no indication of the quality of the tutorial. 

I hope this helps. 

Also, if you need a shape layer to get the benefits of it's options:

Create a shape using the pen tool, select it's "Path" property, and paste.

The simpleset way is: Create a new solid with the composition size, paste the path while selecting that solid: a mask will be added to that solid, apply a "stroke" effect.

What was happening with you: when you paste from Illustrator, a mask was created on that shape layer instead of a shape.