The most important is Blue Cross Cuticle Remover, which is $4.99 with card at Sally's, for the huge bottle! It is a great product, it softens the cuticle to the point where it is very easy to remove, and also moisturizes the skin around the nail. To push up and remove the cuticle, I use the All Seasons Spoon Pusher & Pterygium Remover, which is $9.99 with card at Sally's. I think it is meant for toes, but it works well for my fingers (though the spoon end is rather large for my nails beds). I will explain this one a bit more.
First let me show you my nails before I remove the cuticle, they don't look terrible but there is a tad bit of cuticle growing onto the nail plate, which just looks like roughness on the nail:
The first step is using the Blue Cross cuticle remover. I have decanted some into an empty Gelous bottle, it's much easier to apply with the brush (I promptly spilled this bottle on my rug right after this photo):
The above photo shows my cuticles immediately after swiping the remover around the cuticle. I also brush it under the free edge of my nail, to help remove any excess skin and roughness. The next photo shows my nails right after I push the cuticle back from my nail plate, using the spoon end of my tool.
This next photo is kind of icky, be warned! I next use the Pterygium removing end of the tool, which has a point attached to a curved scoop, if that makes sense. It is perfect for "scooping" under the eponychium and skin to remove pull the excess cuticle out. You must be gentle! I imagine it is very easy to damage the nail plate (I believe it's called the matrix? the part that is under your eponychium) because of the sharp point of the tool. Anyways, here is some of the dead, softened cuticle that's been pushed and detached:
Pretty gross, right?? Yeah, you may not notice it in my first photo, but that stuff will only worsen the longer between times you remove it! I try to remove my dead cuticle once a week, and I notice a difference in how smoothly my polish applies and that it's easier to not flood the cuticle, because my nail plate is smooth!
**please keep in mind I am NOT a professional, this is simply the method I use for cuticle removing and nail care.