There are simple requirements to convert to Islam; you simply have to say the Shahada (the below is the Sunni version, which is the one usually quoted):
لَا إِلٰهَ إِلَّا ٱلله مُحَمَّدٌ رَسُولُ ٱلله
lā ʾilāha ʾillā llāh muḥammadun rasūlu llāh
IPA: [laː ʔɪˈlaːha ˈʔɪl.lɑɫˈɫɑː mʊħamˈmadʊn raˈsuːlʊlˈɫɑː]
There is no god but God. Muhammad is the messenger of God.
Then confirm and declare yourself Muslim and follow the tenants of Islam.
The process does not require any witnesses, you can one day - sitting at home, decide to become Muslim, say the Shahada and follow the tenets of Islam.
Practically, however, such undertaking is done after consultation with an elder or Islamic religious scholar, usually done in front of witnesses (i.e. declared in public - but again, its not a requirement). Most often it is done after the weekly Friday prayers; however again there is no such strict requirement.
Since the process is so simple - for the purposes of admission to the Holy Cities, a certificate is required if you are convert and not born a Muslim. This certificate simply notes down the details of the date, time, the particulars of the witnesses and the certificate is usually signed by the Islamic Center (usually, the prominent Mosque in the area) administration; most often by the director or the designated Imam of the Mosque.
There are no hard and fast requirements, but the certificate serves as "proof" (for legal purposes) that you have converted to Islam.
