If I understand correctly, negative rights oblige inaction and positive rights oblige action.
You have a negative right to not be killed--someone can't hold your head underwater and drown you, for example. But what if someone walks by and sees you drowning? Do you have a positive right to be kept alive--is that person obliged to jump in and try to save you? If you are left to drown your negative right isn't being violated--no one is killing you. However your positive right (if you have one) is being violated.
Freedom of speech is a negative right--you have the right to not be silenced. If freedom of speech were a positive right, maybe it would be more like the right to have people listen to you. If you say whatever you want but no one listens to you, your negative right is not being violated though your positive right (if you had one) would be.
Right to counsel is a positive right--you have the right to be represented by an attorney, including having counsel appointed if you can't afford it yourself. If right to counsel were a negative right, maybe it would be more like the right to not be stopped from hiring an attorney. Again, notice the difference: if you can't afford an attorney, the negative right isn't being violated. No one is stopping you from hiring an attorney, you just can't afford one. But if you can't afford an attorney, your positive right is being violated--the right to be represented by counsel.
In summary, I guess negative rights basically mean "You leave me alone as long as I'm leaving you alone" and positive rights basically mean "You have to provide me something."
It seems like people increasingly think of rights as positive rights. Rights to education and health care, rights to information (see California's recently defeated "Right To Know" proposition), rights to retirement income, and so forth.
I think any society needs a bit of both negative and positive rights, but if we had to pick one type or another I'd lean towards "you leave me alone and I'll leave you alone" over "you owe me stuff and I owe you stuff."