Note: you can see Article 15 of the TEU for the European Council, Article 19 for the Court of Justice, and different sections of the TFEU for the European Central Bank and the Court of Auditors. It's fairly straightforward since it mostly requires a representative by member plus a president (or other similar positions). As so I'll focus on this answer on the EU parliament which has more complex (and "flexible") rules.
The allocation (apportionment) of seats in the EU parliament is decided by treaty following the rules exposed in Article 14 of the Treaty on European Union. Furthermore the actual composition is decided by the European Council.
The European Parliament shall, jointly with the Council, exercise legislative and budgetary functions. It shall exercise functions of
political control and consultation as laid down in the Treaties. It
shall elect the President of the Commission.
The European Parliament shall be composed of representatives of the Union's citizens. They shall not exceed seven hundred and fifty in
number, plus the President. Representation of citizens shall be
degressively proportional, with a minimum threshold of six members per
Member State. No Member State shall be allocated more than ninety-six
seats. The European Council shall adopt by unanimity, on the
initiative of the European Parliament and with its consent, a decision
establishing the composition of the European Parliament, respecting
the principles referred to in the first subparagraph.
The members of the European Parliament shall be elected for a term of five years by direct universal suffrage in a free and secret
The European Parliament shall elect its President and its officers from among its members.
To give a more practical example. Last year the European Council decided on the new rules of the parliament composition after the UK left the EU. The rules they choose are these:
In the application of Article 14(2) TEU, the following principles
shall be respected:
– the allocation of seats in the European Parliament is to fully
utilise the minimum and maximum thresholds per Member State set by the
TEU in order to reflect as closely as possible the sizes of the
respective populations of the Member States,
– degressive proportionality is to be defined as follows: the ratio
between the population and the number of seats of each Member State
before rounding to whole numbers is to vary in relation to their
respective populations in such a way that each Member of the European
Parliament from a more populous Member State represents more citizens
than each Member of the European Parliament from a less populous
Member State and, conversely, that the larger the population of a
Member State, the greater its entitlement to a large number of seats
in the European Parliament,
– the allocation of seats in the European Parliament is to reflect
demographic developments in the Member States
Just as a curiosity this would be the new allocation of seats in the EU parliament.
NOTE: this will likely not be followed considering the latest developments on Brexit. It's in this answer for illustration purposes only.
EDIT: In the web page EU institutions and bodies in brief you'll find the list of the several institutions within the EU. Some of those only have sparse influence on policy. I will not describe the selection process for all of those bodies but that document should put you on the right track should you wish to do so. Most of this answer will focus instead on the EU parliament with pointers to other EU major institutions.
In any case the list of bodies and institutions is the following:
Council of the European Union
Court of Justice of the European Union (CJEU)
European Central Bank (ECB)
European Court of Auditors (ECA)
European External Action Service (EEAS)
European Economic and Social Committee (EESC)
European Committee of the Regions (CoR)
European Investment Bank (EIB)
European Data Protection Supervisor (EDPS)
For legislation (which I believe are the more relevant to your question since it directly relates to policy):
Law-making There are 3 main institutions involved in EU legislation:
the European Parliament, which represents the EU’s citizens and is directly elected by them;
the Council of the European Union, which represents the governments of the individual member countries. The Presidency of the Council is
shared by the member states on a rotating basis.
the European Commission, which represents the interests of the Union as a whole.
Two other institutions play vital roles:
The EU has a number of other institutions and interinstitutional
bodies that play specialised roles:
the European Central Bank is responsible for European monetary policy
the European External Action Service (EEAS) assists the High Representative of the Union for Foreign Affairs and Security Policy,
currently Federica Mogherini. She chairs the Foreign Affairs Council
and conducts the common foreign and security policy, also ensuring the
consistency and coordination of the EU's external action.
the European Economic and Social Committee represents civil society, employers and employees
the European Committee of the Regions represents regional and local authorities
the European Investment Bank finances EU investment projects and helps small businesses through the European Investment Fund
the European Ombudsman investigates complaints about maladministration by EU institutions and bodies
the European Data Protection Supervisor safeguards the privacy of people’s personal data
the Publications Office publishes information about the EU
the European Personnel Selection Office recruits staff for the EU institutions and other bodies
the European School of Administration provides training in specific areas for members of EU staff
a host of specialised agencies and decentralised bodies handle a range of technical, scientific and management tasks
Furthermore you have positions related to smaller offices, or within EU political parties (which keep coming and going and almost certainly chose their own rules of selection).
EDIT (3, May, 2019): Regarding the influence of different members in the EU
Today a study reported by EUobserver and made by a Danish Europe Think Tank analyses the influence of EU members. The paper is named WHO IS BIG IN BRUSSELS?:
RESUME How come some small member states in the EU have more influence over
policy-making than their size suggests they should have? This informal fact of EUcooperation is repeatedly insinuated by expert observers, surveys and scorecards,
such as those by the European Council on Foreign Relations. Here, Sweden and the Netherlands, for instance, are believed to have more influence than more populous
countries such as Italy or Romania. While the Franco-German duo may be the EU’s
undisputed powerhouse, there is no linear relationship between size and influence
over EU policy among the smaller member states.
I cannot fully subscribe to their methodology (the name of the paper might be bigger in scope than the study) but there are some interesting pointers. For example, and although France and Germany are by far the largest members, the staff number does not correlate totally with size with some small countries having large representations:
Moreover they also built a map of the countries they consider over-represented, and under-represented:
Last but not least they make an analysis about the capacity of different members doing coalitions to achieve some objective in the EU:
The ECFR finds that Germany and France are more effective at building coalitions than any other member state. While they are assisted by their great size in sustaining a widespread perception within the Union that they are the most important partners in integration initiatives, the think tank also notes that France and Germany’s coalition building success is due to their uniquely high levels of interaction with their EU allies. This is a variable that is linked to the availability of resources at the perm-reps, and here Germany and France are by far the biggest in our survey.
In a similar vein we may speculate that staff numbers also contribute to explain the ECFR’s
findings that, among the Nordic trio in the EU, Denmark, Finland and Sweden, Sweden contacts Finland much more ‘on European policy matters’ than it contacts Denmark, and Finland contacts Sweden, the Netherlands and Estonia more than it contacts Denmark. While Denmark’s influence may be lower than its neighbours due to its EU-opt-outs (on justice and home affairs, defence and the euro), it may also be part of the story that out of these countries, the Danish permrep is the only one to rank below its population rank. At least, its relatively small permanent representation, and short duration of secondments from most home ministries, do nothing to alleviate the possible loss of influence from the opt-outs that Denmark may be experiencing. In this respect, it is noteworthy from our figures that especially Slovenia appears to try to compensate for its small mission in Brussels by prioritising the presence of home ministry staff and by ensuring a longer duration of secondments.