Although a dynamic vocal mic such as a Shure SM58 will do the trick, a capacitor mic is more sensitive, especially to high-end frequencies. A good example of an affordable capacitor mic is the Rode NT1. Unless you have a studio with a separated-off vocal booth, avoid omni-directional mics, as these could pick up unwanted sounds in the room as well as the vocal in front of them. The mic should be mounted on a stand, not held in the singer's hand.
Instruct the singer to maintain a consistent position in front of the microphone, about 6 to 8 inches away. She must not move around while recording, as this will mess up your levels. Fit a pop shield to the mic stand, to minimize the plosive, "pop" sound generated when singers enunciate the letter "P." This has the added benefit of keeping distance between the singer and the mic. It's normally a good idea to keep working your singer until you have at least three acceptable takes, as the second- and third-choice takes could come in handy if you choose to double-track the vocal. If you have a song in which the singer intends to change radically from soft to loud, record the soft and loud passages separately, if possible. It will make life easier for you when it comes to mixing.
Once the vocal take is down, play it back on its own to check for any sonic glitches or mistakes that you or the vocalist might have missed. Once you have agreed on the suitability of the take, introduce it to the instrumental track at a level where it is generally prominent throughout. A common problem is consistency of level, so you will probably need to automate the volume up or down at key points. Compression is usually added to stabilize the level of the vocal, picking it up when it drops, whilst preventing it from becoming too loud when it peaks. However, care should be exercised, as too much compression can crush the liveliness out of the vocal performance.
Manipulating the equalization ("EQ") of your vocal will help it to "sit" in the mix, while allowing the instruments to shine too. A high-pass ("low-cut") filter is often added at around 80Hz, to remove all sub-bass frequencies. Usually, complementary EQing of instruments can also be helpful. If, for example, you determine that you want your vocal to come through strongly in its natural frequency range -- normally this will be at around 2KHz -- try cutting the EQ of your guitars or keyboards by about 3dB at this frequency to create more room.