there are some input fields that are required, if they are not showing on the CSV, then they won-t show in red.
One of those is the name of the contact.

I'd suggest doing the following. 
send a screen of the CSV you are using so we see the fields you are filling. 
and a screen of the error so we can better guide you =)

using the template is a good idea, but sometimes, you make some mistakes like missing a required field or leaving something blank where it shouldnt. 

What i tend to do is, Create a full Contact with fake info, but fill in everything in the contact form. 
then, click it on contact lists, and go to actions -> export. and make sure to click on "I'll use it later" option so i get all the info i need when i decide to import it. 
after this, try to match all the fields from this file with the information you wnat to import. 

be ready and expect to have several errors the first times... its normal. 

for example, if you try to import companies and contacts from said company at the same time, you tend to create problems when importing.