When asked for your name, context will determine whether just your first name or your full name, your nombre completo, is expected.
Hispanic names often include a first name and a middle name, which is called a segundo nombre. Normally two last names are used: first the apellido paterno, then the apellido materno. If only one is quoted, it will normally be the paterno.
In anglo cultures, the mother’s maiden name may be somewhat confidential, being asked for to confirm your identity during phone transactions. You can tell native speakers that you don’t have an apellido materno, for example, if it does not appear on your official documents as part of your name. However, some computer systems require something to be entered in that field. Native speakers unfamiliar with English names may need some guidance in parsing your full name into nombres and apellidos.
Should You Hispanicize Your Name?
On the first day of high school Spanish, we were assigned names in Spanish. Mark became Marco. It was considered part of the cultural journey we were about to take. For various reasons, this is now considered unfashionable in language teaching.
You may be considering using a Spanish name, perhaps one similar to your English name, with native speakers. Doing so has the advantage of being easier for them to pronounce and understand. And some people will call you by the Spanish version of your name, regardless of how you introduce yourself initially. Using a Spanish name may make you may feel more integrated into the culture.
However, some native speakers may consider your use of different names in Spanish and English to be humorous, off-putting or even insulting. You’ll be asked what your real name is, and they may insist on using it in Spanish. So now some people are calling you by your Spanish name and others by your English name, and you’ll forever be explaining why.
For these reasons, as a general rule I recommend keeping your English name when communicating in Spanish. You can’t force your way into a culture. The acceptance has to be mutual.