I'm curious how you would handle following Database access.
Let's suggest you have a Computer which Hosts your database as part of his server work and multiple client PC's which has some client-side-software on it that need to get information from this database
AFAIK there are 2 way's to do this
so what i like to know is: What are the pro and contra's of each solution? And are other solutions out there which maybe "better" to do this work
What you have described sounds like two different kind of multitier architectures. The first point matches with a two-tier and the second one could be a three-tier.
AFAIK there are 2 way's to do this
You can divide your application in several physical tiers, therefore, you will find more cases suitable to this architecture (n-tier) than the described above.
What are the pro and contra's of each solution?
Usually the motivation for splitting your application in tiers is to achieve some kind of non-functional requirements (maintainability, availability, security, etc.), the problem is that when you add extra tiers you also add complexity,e.g.: your app components need to communicate with each other and this is more difficult when they are distributed among several machines.
And are other solutions out there which maybe "better" to do this work.
I'm not sure what you mean with "work" here, but notice that you don't need to add extra tiers to access a database. If you have a desktop application installed in a few machines a classical client/server (two-tier) model should be enough. However, a web-based application needs an extra tier for interacting with the browser. In this case the database access is not the motivation for adding this extra tier.