Bell - The part of the trumpet where the sound comes out of. Mostly made of brass and can either be lacquered in gold, which produces a more mellow sound, and silver-plated, which produces a brighter sound. Other trumpet manufacturers create specially made bells such as those made of sterling silver. Alterations to the bell of the trumpet affects its sound. The size of the bell, otherwise known as the flare, also affects its sound. Smaller bell flares sound sharper while bigger flares sound mellower. Higher-end trumpets uses tuning bells which are removable. The musician can alter the sound by adjusting the tuning bell.
Finger Hook - It enables the other hand of the player to be free and perform other musical tasks.
Valve Casings - It is also referred to as the three cylinders that is attached to the pistons.
Piston - Also called valve pistons. When you blow into a trumpet's mouthpiece, the pistons reroutes the air path into different slides. There are three pistons: the first one is the one closest to you, the second is in the middle and the third is the farthest. These three pistons aren't interchangeable, so you should take note of their proper positions when aligning them.
Leadpipe - The tube from the mouthpiece to the tuning slide.
Tuning Slide - The part of the trumpet which can be pulled or push to adjust the tune.
Valve Slides - It helps the trumpet produce sound as well as adjust the pitch of notes. There are three valve slides: the first slide lowers the highest note a whole step (also called a fundamental, which is produced when you're not holding down any valve), the second slide lowers it a half step and the third slide is commonly used to produce notes that are lower in register.
Mouthpiece - This is the part where the player places his lips and blows into. Mouthpieces are made in varying sizes and different materials such as brass.
Here is an illustration of the parts of a trumpet from Trumpeter's Fanfare to guide you accordingly.