Absolute power of a signal is measure in wattage. The bel or decibel system can only describe relative power- a gain of 3 dB means your signal is 2 times as strong as it was before, but the dB scale doesn't define where you're starting from or what your 'zero' is.
dBm, dBi, and dBd are different ways to define what your zero is. So, signal power in dBm means that you're using a logarithmic (bel-based) scale relative to 1 mW of power; dBi means you're using an isotropic antenna as your base; dBd means that you're using a dipole antenna as your base.
The reason you see negative values is that you're representing small but positive numbers, on a logarithmic scale. In logarithms, the value indicated represents an exponent... for example, under a log 10 scale, a value of -2 represents 10 to the -2 power, which equals 0.01. Likewise, a negative dBm means that you're applying a negative exponent in your power calculations; 0 dBm equals 1 mW of power, so -10 dBm equates to 0.1 mW, -20 dBm equates to 0.01 mW, and so forth.
dBm is used to specify transmit power or receive sensitivity, since the 'm' refers to power. dBi and dBd are used to specify how much more focused one antenna is relative to another. If you imagine radio coverage as a balloon, changing antennas allows you to squish the balloon in some directions to cause it to expand in others; what antennas do is to take your existing radio energy and change the shape of your coverage area without increasing your total coverage volume.
An isotropic antenna, the base antenna for the dBi measurement, doesn't exist; it's a theoretical antenna with a perfectly spherical coverage pattern. So, if you see an antenna rated in dBi, its being described as how focused the coverage area is relative to a sphere. A low dBi rating means that it's pretty spherical; higher numbers mean that the coverage area is more deformed.
The dipole antenna is the lowest-gain real-world antenna- providing about 2.15 gain above the perfect sphere described above. It's still an omnidirectional antenna; the gain comes because the vertical coverage is foreshortened and the horizontal coverage is slightly expanded; imagine pushing in on the top and bottom of a balloon to get a slightly more disk-like shape. Some people use dBd instead of dBi since the dipole is the simplest real-world antenna; to convert dBd to dBi just add 2.15... i.e. a 3.0 dBi antenna could just as easily be called a 0.85 dBd antenna.
Note that there's no 1-to-1 correlation between dBm and dBi/dBd: dBm measures power, dBi measures an antenna's signal-shaping characteristics ("gain").
