Contact/Relay -- Relays are electronically controlled switches that activate the various components of your HVAC system. They manage everything from the power going to the motor to automatic dampers, humidifiers, etc. Most of them are controlled by the thermostat. The most common failure for a relay is being stuck in the "open" position. Separated from its assigned contact point, it fails to complete the connection and send the message to whatever it was supposed to operate. This failure usually occurs from use over time. Each time a relay connects and sends its signal, the electrical arcing from point to point eventually causes wear and tear.

Indoor Fan Motor -- Indoor fans circulate the air from your house and through the system. Age and dust buildup are the two biggest contributing factors to their failure. When they are starting to fail they will often make noise. Failure of the fan will result in no heating or cooling, and if it is not repaired soon, it can cause other components to fail.
I live in a 1950's ranch house in the South. The prior owners were smokers. Based on recommendations from one of the local conservation organizations, I decided to have a home energy audit done. As part of their findings, they recommended additional insulation in the attic and duct cleaning and sealing. This turned out to be a very problematic process. In terms of the ducts, it was necessary to remove the metal grates. However, these grates had been painted over many times and shoe molding covered the bottom of the grates. The vendor was not able to remove the grates without damaging them in the process. Because these grates dated back to the 1950's, it was difficult to find appropriate replacements. The ones I finally discovered were correct except they were 1/4" wider. This meant that the shoe molding had to be carved out. The duct cleaner could not provide any assistance with this problem. It took me a month of long evenings replacing the vent covers myself, carefully chiseling out the shoe molding and repainting. In terms of the extra installation in the attic, the same vendor used blow in insulation. Unfortunately, they did not recognize that the 13 floodlights whose mounting extended into the attic were not designed to be buried under insulation. The problem manifested itself in terms of expensive flood light bulbs failing continually. So, I had to have all of the light housings replaced. Yet another unexpected expense. Despite the expense, pain and suffering, I cannot report any improvement in my heating costs or the "quality" of the air. To be fair, I do not have any allergies and did not undertake this for air quality improvements but rather heating efficiency improvements.

I have a white gritty dust throughout my whole house. Dust and it comes right back. We have central air. In 2013 had all the air conditioning tubing in the attic replaced. Old ones were completely clean. Why?? Thought dust would be in there. Put everything in plastic containers. Furniture everything is always covered with this. Problem is always there year round. Furnace people said it is not coming from furnace. We built house in 1971. Only started with this @ 19 years ago. Thinking when air conditioning went in. We cough all the time and sneeze. I do not have friends come here because I hate how dirty my house looks.
Many central air conditioning systems use the furnace blower to distribute cool air through the home. If you do not have a central heating system installed, it is cost-effective to install a heating and air conditioning system at the same time. If you already have central heat, you can use the existing fans and duct system for the central air system.
Most organizations concerned with duct cleaning, including EPA, NADCA, NAIMA and the Sheet Metal and Air Conditioning Contractors' National Association (SMACNA) do not currently recommend the routine use of sealants to encapsulate contaminants in any type of duct. Instances when the use of sealants to encapsulate the duct surfaces may be appropriate include the repair of damaged fiber glass insulation or when combating fire damage within ducts. Sealants should never be used on wet duct liner, to cover actively growing mold, or to cover debris in the ducts, and should only be applied after cleaning according to NADCA or other appropriate guidelines or standards.

You should be aware that although a substance may look like mold, a positive determination of whether it is mold or not can be made only by an expert and may require laboratory analysis for final confirmation. For about $50, some microbiology laboratories can tell you whether a sample sent to them on a clear strip of sticky household tape is mold or simply a substance that resembles it.