The issue is a little bit more complex than what the question specifically addresses.
NCBI is indeed a "resource", that it means you can have specific informations contained in databases (collection of data organised "as a set of formally described tables") and provided by servers (physical or logical entities that perform calculations or functions needed by applications, such as databases).
EMBL is an institution composed by several labs.: surely has resources, therefore surely has databases that "work" thanks to (or are executed by) servers.
A Database is a collection of related information that may not even involve computers at all. However, the database in the computer sense is stored on a disk and plays the role of an information repository.
A Database server is the process that makes the data in the database available to the outside world.
Server is a place that holds tools, pipelines, databases and many other applications where some times users or guests could log in and use them. There could be servers for many different requirements some of them only holds information and facilitate information transfer, but does not allow users to have direct interactions, where as others allow users to have direct interactions. So one could find database servers, mail servers, file servers, web servers and so on. Servers mainly serve a request made by a tool, another machine or a user. Hope that this clarifies what your doubts.
In analogy to driver we can assume server. Consider a local LAN model where many clients (USERS) generates multiple queries and all queries must have to be redirected through a common gateway modem. To control the traffic we may place another better capacity machine in between user and modem and we may regard it as a local server which is serving for your local end queries. Similar model is implementable at a higher level i.e. NCBI where you have no direct connection with their database but you can access all the different databases through a common address. It simply accept your query and redirect it to the concerned then in return reflects the formatted readable results to end user (YOU). Database is just a ordered collection of relevant information while server may serve you for a number of different queries.
Digital databases are collections of data, for example personal information, wages records, medical records, genomic data, transcriptomic data, proteomic data, stored based on a schema (logical structure) where queries are served based on the schema. There are different database engines to design databases and store data. Some of major engines are mysql, postgresql SQLite. There are many others and many other tools for efficient maintenance and handling of databases effectively. When these engines are used engine specific drivers are needed for queries using other tools. These drivers serves as bridges between database engines and other tools (programming languages for examples) for bidirectional communication. Mostly used database language is called structural query language (sql).
I really don't know whether you could call NCBI or EMBL database or not. However, my feeling is that they are servers that facilitate querying collection of databases based on default or user selected options. NCBI provides many different front end interfaces for users to interact with databases of interest. So is EMBL I might be wrong.
I agreed with Dung LE, NCBI and EMBL is both server as well as Database. It is consists of collection of various databases at the same time it holds various tools and online server for data analysis. So, if you are talking about the whole thing inside the NCBI and i feel NCBI is a website which consists Databases and online server for data analysis tool such as BLAST etc.
I would say that neither NCBI nor EMBL are either a database, or a server. Those initials refer to institutions, each of which hosts numerous databases, which in turn reside on servers maintained by the staff at those institutions.
GenBank is a database hosted by NCBI, as is RefSeq or UniGene. The genomes in Ensembl are a database (or some might prefer, a meta-database, since each genome can be a standalone database), one of the developers/hosts of which is EMBL-EBI.
Databases are just collections of things, and the connections between them.
The issue is a little bit more complex than what the question specifically addresses.
NCBI is indeed a "resource", that it means you can have specific informations contained in databases (collection of data organised "as a set of formally described tables") and provided by servers (physical or logical entities that perform calculations or functions needed by applications, such as databases).
EMBL is an institution composed by several labs.: surely has resources, therefore surely has databases that "work" thanks to (or are executed by) servers.
both NCBI and EMBL should be considered as COMPOSITE DATABASE, something like collection of collections. though it hosts many tools, they are organized as a data itself to analyze the primary source.