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You really need to read the help and instructions that came with your cad program.
The difference between a part and a component is:
The part could be a "10K 0805 resistor".
The "10K 0805 resistor" part links a resistor symbol, an 0805 component, pin mapping and other information all together.
The component could be "an 0805" footprint. It is only a footprint (pads) of the component.
The part may be used several times within the design, so you may have 22 times the "10K 0805 resistor".
If your design also had 5 times "1K 0805 resistor".
The net is the complete connection between all connected pins, i.e. all pins that goto GND connect to the GND net.
If your circuit has an input terminal that connects to one end of a fuse, that connection between those 2 points is called a net.
The other side of the fuse, it connects the fuse to a regulator via a reverse polarity diode, this is another net.
Is "wire" a term your package uses in the schematic or the PCB? and what is the package?
You would have 2 parts:
"10K 0805 resistor"
"1K 0805 resistor"
But 27 components.
Because the part is used multiple times within the design.
Looks like it's DipTrace you're using. They have a tutorial, it's pretty easy to search for keywords, like I do.
Net is the connection between two, three, four or more pins. Wire is like a segment of the net. Because sometimes net is build from several wires to connect several pins.
Component is a schematic symbol and a PCB footprint together. (I'm a DipTrace user) They often say Component about schematic symbols.
Part. Some schematic symbols are built of several separate parts, so called multi-part components, that's what you see in that panel on the picture.