Library is a set of components which I can use within my schematic. Libraries can be imported and used or if required we can also create our own library

My very first component and which is ATTINY44 we have imported from fab library and within schematic view

That green line is called as a net. Which connects one component to another component. Its labelled as VCC. To label it do a right click and select Name. Provide a name and then again right click and select label. Place a label appropriately

Net with same labels virtually connects with each other. For example in this case VCC of ATTINY44 will connect with VCC of USB. This is how we can connect using references. Using this technique we can make this design modular and move a part of it to another page to make it neat and clean

Donwloaded png image for Hello ftdi board for reference

Imported components required for hello ftdi board

• ATTINY44
• 6 pin connector
• LED
• Capacitor
• Resonator
• Resistors

Auto routing creates traces algorithmically and try to connect all of the traces to complete this circuit

Electronic cad software helps us to make PCB layouts which suits our logic. But to manufacture PCB particularly on milling machine we need to take care of clearances.
For example if milling bit is of size 0.5mm then traces should be defined with atleast 0.5mm clerance. Otherwise you may end up with very thin traces or else sometimes traces will vanish
Design rules can be set in eagle cad under menu Tools > DRC

After clicking on Check button it will check for areas where design rule is getting violated. It will highlight area with red hashed markings. You need to resolve these issues. After you are done with this you are ready for milling your board.
Auto routing will also take care of these design rules you have set and will always follow design rules. You may not get 100% autorouting success when you set design rules.