That is an Eastern Red Cedar which will reach a mature height of 30'. If you are looking for privacy screens I would also recommend that you look at: Nellie R Stevens Hollies, Eagleston Hollies, Oak Leaf Hollies, Wax Leaf Ligustrum, Brodie Easter Red Cedar, Spartan Juniper, Cherry Laurel, Little Gem Magnolias & Bracken Brown Beauty Magnolias.

how far apart do you have to plant these?

That really depends on what you are looking for. For privacy needs it depends on how immediate your privacy needs are. You can buy several and plant them really close together to give more instant privacy, or you can plant them farther apart allowing for them to grow in over time. If they are going in your flower beds you can plant them a few feet away from the house.