No app required. To do this save contacts as 'Google contacts'. They will be automatically saved to your Google (Gmail) account 'in the cloud' and will sync (download) to a new phone when you log into your account with it.
Note that only Google contacts sync. SIM contacts or any saved as 'Phone' contacts do not sync. Those must be exported then imported to your Google account. If transferring contacts from a non-Android phone, export them as a .vcf file, import that to your Gmail contacts and they are ready to sync to the new phone.