To craft custom weapons you must dock and enter a station, then click on Weapon Lab.
But first you need several ingredients. Metal, Machinery, Electrical Units and Fusion Cells, and some cash.
Metal can be mined from asteroids and planets.
Machinery can be created out of Metal by the use of a constructor station.
Electrical Units can be created out of Platinum by the use of a constructor station. Platinum can be mined from asteroids and planets.
Fusion Cells can be created out of Hydrogen by the use of a constructor station (5 to 4 ratio). Hydrogen can be harvested from nebulae or a sun's corona if no Fuel Converter is equipped.
For the amount required for each type of weapon see below at the list of the different weapon classes.

Yield is the amount of damage the weapon does per shot.
Heat is the amount of heat generated per shot. Heat makes it easier for missles to lock onto you.
Energy is the amount of Energy drained from your weapons power per shot.
Range is the distance at which your weapon will be able to lock onto an enemy and damage them.
Rate is the amount of shots fired in one minute (RPM), and the cooldown time after firing a shot in milliseconds (C).

Emitter Output increases your weapons yield, at the cost of slower firing rate and more energy drain.
Actuator Speed increases firing rate at the cost of yield.
Capacitor Reserve increases yield and range at the cost of firing rate and more energy drain.
Heatsinks increase your yield and firing rate by cooling your weapon system, as weapons will slow down firing to prevent overheating.

While higher tier weapon classes look much better at default settings, changing them to the same setting and comparing reveals them to not be that much different than their lower tier counterparts.
Their damage per shot and range increases, firing rate decreases but energy drain increases.
This is the case across the whole range of settings.
For simplicity's sake I have set all sliders to 100 in the following examples.
Note that this isn't necessarily the best to actually use, considering that each setting also has drawbacks.

Plasma
Requirements: 10xMETL, 7xMACH, 10xELEC, 1xFUSN, $16200
Yield: 67
Energy: 16
Range: 760
Rate: 417 RPM / 12 C
Heat: 164
Colour: red

Metal
Requirements: 16xMETL, 12xMACH, 15xELEC, 2xFUSN, $16200
Yield: 72
Energy: 17
Range: 770
Rate: 407 RPM / 13 C
Heat: 168
Colour: blue, with orange trail

Rail
Requirements: 19xMETL, 17xMACH, 20xELEC, 3xFUSN, $16200
Yield: 77
Energy: 21
Range: 780
Rate: 400 RPM / 14 C
Heat: 172
Colour: purple, with blue trail

Fusion
Requirements: 22xMETL, 22xMACH, 25xELEC, 4xFUSN, $16200
Yield: 82
Energy: 31
Range: 790
Rate: 390 RPM / 16 C
Heat: 176
Colour: white, with blue trail and violet glow

Emitter Output increases your weapons yield, at the cost more energy drain.
Your Laser Beams will match the distance of your current Particle Cannon.
Again for simplicity's sake I have set all sliders to 100 in the following examples.

Refractor
Requirements: 6xMETL, 3xMACH, 5xELEC, 1xFUSN, $16200
Yield: 47
Energy: 10
Heat: 122
Colour: cyan

Metal-Vapor
Requirements: 8xMETL, 6xMACH, 7xELEC, 2xFUSN, $16200
Yield: 51
Energy: 11
Heat: 124
Colour: green

Coil
Requirements: 9xMETL, 8xMACH, 10xELEC, 3xFUSN, $16200
Yield: 55
Energy: 12
Heat: 126
Colour: magneta

Neodymium
Requirements: 11xMETL, 11xMACH, 12xELEC, 4xFUSN, $16200
Yield: 58
Energy: 14
Heat: 128
Colour: red

Fusion
Requirements: 12xMETL, 13xMACH, 15xELEC, 5xFUSN, $16200
Yield: 62
Energy: 20
Heat: 130
Colour: blue

Equipment can be created at constructor stations from various items.
The type of item determines what kind of Equipment you can craft.
The amount of items determines the level of it.

Fulcrum Drives C5 = 25x Fusion Cells = 29x Hydrogen
Cargo Scanner C5 = 25x Electrical Supplies = 25x Platinum
Repair System C3 = 25x Machinery Parts = 25x Metal
Fuel Converter = 15x Anti-Matter