Spark is a distributed in memory processing engine. It does not need to be paired with Hadoop, but since Hadoop is one of the most popular big data processing tools, Spark is designed to work well in that environment. For example, Hadoop uses the HDFS (Hadoop Distributed File System) to store its data, so Spark is able to read data from HDFS, and to save results in HDFS.
For speed, Spark keeps its data sets in memory. It will typically start a job by loading data from durable storage, such as HDFS, Hbase, a Cassandra database, etc. Once loaded into memory, Spark can run many transformations on the data set to calculate a desired result. The final result is then typically written back to durable storage.
In terms of it being an alternative to Hadoop, it can be much faster than Hadoop at certain operations. For example a multi-pass map reduce operation can be dramatically faster in Spark than with Hadoop map reduce since most of the disk I/O of Hadoop is avoided. Spark can read data formatted for Apache Hive, so Spark SQL can be much faster than using HQL (Hive Query Language).
Cassandra has its own native query language called CQL (Cassandra Query Language), but it is a small subset of full SQL and is quite poor for things like aggregation and ad hoc queries. So when Spark is paired with Cassandra, it offers a more feature rich query language and allows you to do data analytics that native CQL doesn't provide.
Another use case for Spark is for stream processing. Spark can be set up to ingest incoming real time data and process it in micro-batches, and then save the result to durable storage, such as HDFS, Cassandra, etc.
So spark is really a standalone in memory system that can be paired with many different distributed databases and file systems to add performance, a more complete SQL implementation, and features they may lack such a stream processing.
