A lakehouse is an architectural design to build a data warehouse using data lake/big data tools.

So it takes elements from classical data warehousing and skips the whole RDBMS part and inserts other elements like HDFS etc.

So think of it as a data warehouse with modern scale out tools.

Delta lake is a file format which will contain the actual data (think of it as a column store table) based on the Parquet file format, but with added functionality (it is ACID compliant).

How do the two relate?

Well, Delta Lake fits quite well into a lakehouse, especially because delta lake has upsert (MERGE) functionality (that most other file formats lack), change data feed, clones, rollback etc.

It also has several performance enhancing functions which makes it a very suitable file format to build your lakehouse upon.

So when you build a lakehouse, delta lake will probably be part of it (together with parquet and maybe some others).