what is equivalent of a windows service on azure?

You should look at continuously-running web jobs. See Running Background tasks with WebJobs on Microsoft Azure.

Other choices are PaaS cloud services worker roles and Azure Service Fabric reliable services - but these are likely overkill if you just want a basic service.

Azure function is a good candidate for migrating windows services into something cloud based. Azure function can be triggered by a timer and so like a windows service can be scheduled at a certain time of the day for example.
Please give a read to my notes that I wrote while I worked on such a migration: https://dumanhilltechnologies.com/blog/windows-service-migration-to-azure-function/

There is no specific way to run your code in Azure. You have lots of choices, and which you choose is really up to you (and a matter of opinion). But, objectively speaking:

• Install your service as you always have, in a Windows Server VM
• Run your code, without the Windows Service wrapper, in a VM (either Windows or Linux, depending on language)
• Pull your core code out of the service, and run it within a web/worker role (cloud service).
• Run your code in a WebJob.
• Run your code in a Web App (you'd need to add some way to get to it, like a REST API sitting in front of it)

I see that @Neil suggested Service Fabric in his answer. That works too, except you'll need to learn about Service Fabric in general, since it works a bit differently.