Your gutter is likely to last at least 10 years when properly installed and maintained. This is a low estimate and depending on what material you use, you can expect many more years.
Aluminium gutters: 20-30 years
Vinyl gutters: 10-15 years
Steel gutters: 15-20 years (if coated for rust protection)
Copper gutters: Highly durable and can last upwards of 50+ years.
Keep in mind this all assumes proper maintenance and care has been taken of your guttering system including regular cleaning.