If you are here in the UK on a student visa you can get married either in a civil ceremony (usually in a Register Office) or in a religious ceremony in your normal place of worship. You may need to register the marriage if you get married in a religious ceremony before it will be recognised as a lawful marriage in the UK.
Further information on the procedure is available at: www.sheffield.gov.uk/your-city-council/register-office
There is no provision within the immigration rules for a fiancé(e) to enter the UK for the purposes of marriage or for a proposed civil partnership and to then switch to dependant. Your fiancé(e) would need to apply as Marriage visitor and then return home to apply for a Tier 4 dependant visa.
If one of you is British or has permanent residence in the UK please see here.
If you are not married but have been living together for at least two years you may be able to apply to remain in the UK on the basis of this relationship. You would need to be able show documentary evidence of living together such as rental agreements, utility bills, bank statements, letters from employer/university. It will then depend on the immigration status of your partner as to whether you will be able to apply to remain in the UK.
1. I'm in a relationship with a student from outside the EU
Marrying or entering into a civil partnership or living with another non EU international student will not usually affect your existing rights or status. If you want to change your status from student to dependant you will need to check whether you are eligible to do so. Further information on dependant visas is here.
2. I'm in a relationship with a person with a work visa (Tier 1 or Tier 2)
If you marry someone who has permission to be here in any of the worker statuses e.g. Tier 1 or 2, you can apply to become a dependant in the UK provided you are not currently here on a visit visa or a short term student visa.
3. I am in a relationship with an EU national
There are different rules for family members of EEA nationals.
4. I am in a relationship with a British citizen/settled person
New rules were introduced in July 2012 for anyone wanting to stay in the UK on the basis of marriage/CP or co-habitation. You will need to show that you are able to meet a number of criteria including an earned income of £18600 a year (plus additional amounts for any non-British dependant children). For further information please see here.