There are a lot of 'Secretaries' in a Government Department! There are two structures within a typical Government Department.
The Political Structure
The Political Head of a Department is usually called the "Secretary of State for (name of Department)". Sometimes they are known by a different title (The Chancellor of the Exchequer - who has overall responsibility for the Treasury. The Secretary of State for Justice currently has the title of 'Lord Chancellor') - They will be members of the Cabinet.
The next rung down is the Minister of State. At the most junior paid level is the Parliamentary Under-Secretary of State.
Some of the Ministers may be assisted by a PPS (Parliamentary Private Secretary). These are unpaid MPs who acts as a minister's eyes and ears in Parliament, making sure the minister is kept well informed of backbench opinion. While they are not members of the Government, they are regarded as part of the "payroll vote" and are expected to vote with the government - or resign. It is often the first step towards a ministerial career.
Press Here to see the Ministers in the Department of Energy & Climate Change. and Here for the Ministry of Justice.
The Civil Service Structure
In the UK Ministers are assisted by Civil Servants - who are professional staff, not political appointments. There are strict restrictions on their political activity. The most senior Civil Servant in a Department is the Permanent Secretary. They run the civil service within that Department and are also the "Accounting Officer" for the Department. There is an excellent guide to the Civil Service available here. [See about us: Leadership]
Larger Departments may have an official known as the "Second Permanent Secretary". At the next level down are the "Director-Generals". The Structure within DECC is available on a pdf. here.
Ministers have their own Private Office, headed by their Private Secretary. The Private Secretary to the Secretary of State is the Principal Private Secretary.