A very general answer to give you the basics but there will be detailed situations where it is not the complete answer.
A visual approach is one flown all the way down to the runway by the pilot looking out of the window and judging the approach path and descent rate from what he/she sees - there are lighting aids at many airports to help but it may simply be a case of experience telling the pilot whether the aircraft is too high or too low or whatever. It is fairly obvious that the pilot must be able to see where he/she is - although not necessarily be able to see the runway - to make a visual approach.
An ILS is a common type of instrument approach where the pilot uses the instruments on the aircraft and radio signals from the ground or satellites to position the aircraft on the correct approach. Most instrument approaches are intended to get the aircraft to a point where the pilot can complete the landing visually. The radio aids that are available and the aircraft equipment the instrument determine how accurate the approach can be. The more accurate the aids, the lower you will be able to descend before you complete the landing visually (because you have greater confidence about where the aircraft is relative to the runway and any high ground near the airport). The accuracy is indicated by the category of the approach - a cat I approach will be able to bring an aircraft down to 200ft after which the pilot will land visually, at cat III approach will bring the aircraft down to 50ft or less. A cat III approach will be flown by autopilots and and may be approved for what are often called blind landings where, in some cases, the pilots will not have to see anything before landing - although in this case it's usually too foggy to be able to taxi off the runway. Of course, the poorer the weather that the aircarft is permitted to land in, the more training the crew will get and the more back-up systems are required on the aircraft and on the ground etc.
You will also hear other instrument approaches referred to - e.g. NDB or ADF, VOR and, if you're old like me, you might even remember VDF or QGH approaches. All of these are less accurate than an ILS approach.
There are other types of approach in use or under development like MLS, RNAV and GPS - if you Google these terms you'll find more info but they are less common than ILS or NDB approaches.
Hope this helps a bit.