The conventional feature that comes to mind immediately to support the functions you describe is user profile. Conventionality means recognition and familiarity.
Presumably an accounting application has security features that require secure log-in and authentication, as well as security timeout etc. Offer the distinction between business and individual consumer user profiles during this mandatory initial setup, that is, during the "Create user ID" steps on first visit. Embed the profile choice into that setup. I advise against stashing the profile choice into a popup - the modal route (a "popup" dialog) is unnecessary and annoying if your user needs to provide setup information anyway, like user name and credentials.
Does the setup information you're asking your users to provide differ between business and individual users? If one setup requires significantly more data points than the other - more checkmarks to check, fields to fill in, dropdown picker pickings to make - you'll want to place the business versus individual 'roadfork' early into the initial sign-up workflow as the following downstream content of one route will be more elaborate than the other.
The same constraint applies to switching from one profile type to the other while logged on. If the difference is trivial, make it an easy toggle, but if a business user profile is a vastly different beast from an individual one, construct a more 'guardrailed' UI flow by which your user converts from one profile to the other.
To stick with conventional UX (and thus repetition and recognition benefits from other sites), most people familiar with anything requiring log-in / log-out will recognise a user avatar located on your site's header, towards the right, as location of their user profile (somewhat like StackExchange), and expect simple dropdown, show-me-more actions like Settings and Log out when that avatar is clicked.
If the profile change from business to individual or vice versa is easy and does not require significant additional information, that is where you may want to locate it. If that change is fussier and more demanding, I would expect to find the profile switch option under Settings (or "My Profile").
This pattern is easily transferred to a mobile environment, only the arrangement and layout of an otherwise identical information architecture may differ. Exactly how you structure that setup depends on - again - how involved your specific setup steps are. If your profile setup is complex and full of dependencies (content in step #2 depends on step #1), some wizard-style progressive disclosure may be called for - and a wizard with contemporary looking UI need not be a Windows-95-ish-looking dialog anymore these days!
Regarding how obvious the site version should be to you users, how do you plan to market the site? If your business user version is higher end (and possibly paid for, or fee-charging) than the individual consumer version you may want to reflect that in whatever naming convention you apply to your site. Reward your fee-paying customer or subscriber segment by providing a noticeably richer offering whilst encouraging your freebie users to consider upgrading, but without having your site take too obnoxious a salesmanship stance as constant prompts for 'in-app purchases' can have the opposite effect of cheapening your offering.
However if the distinction between business and individual site profiles is immaterial, and the nature and complexity of the profile generation data input required is about the same, I would make this an easy toggle and perhaps add descriptive labelling to the user avatar (see above), below the user name.
It may be worth exploring the business and individual user segments of your planned site via a persona study - even if you already collaborate with a marketing expert. Do you know the extent to which each segment's top-priority needs and expectations differ? Are there differences within each segment? Are there overlaps (e.g. for self-employed individuals where aspects of personal and business finances may merge)? How uniform are the accounting needs of your business segment?
One more thought to the differences between desktop and mobile: Both user segments may engage with one or the other environment for quite distinct reasons. Mobile, on-the-go accounting is immensely practical for instance to track travel expenses - look at the likes of Concur - as you can upload pictures of bills / invoices. But it is probably less usable for more complex workflows like factoring in taxes and such. Consider differentiating exactly what features you offer by environment rather than making uniform desktop and mobile offerings - but be mindful that that sort of sophistication needs to be informed by solid user research. Good luck!
